It's your DM's call, but...
As other answers have stated, Rule 0 says that your DM can say that your warlock doesn't have awareness of how many temporary hit points each casting of false life gives you. Your DM gets to decide how things work in their campaign. Heck, they could even decide that your warlock doesn't know their temp hp total but every other warlock in the world actually does. If that is their solid and reasoned decision, we as answerers cannot dispute that. However, I am going to proceed as if your DM has made this decision based on a misunderstanding, and thus is amenable to persuasion if you can explain their misunderstanding. The first thing to do is ask them why they have decided to rule this way, and then proceed from there.
Do they think this is RAW?
Because it's not. Casting false life again, when you already have one running, gives you a new source of temporary hit points when you already have temp hp. In this case:
Healing can't restore temporary hit points, and they can't be added together. If you have temporary hit points and receive more of them, you decide whether to keep the ones you have or to gain the new ones. For example, if a spell grants you 12 temporary hit points when you already have 10, you can have 12 or 10, not 22.
This passage explains that a new source of temp hp does not add to an old one, but it also explains that when you could gain new temp hp you decide whether to gain the new or keep the old, and it explains this choice in terms of numerical values (12 or 10). You are not just choosing between 'new and old', you are choosing between '12 or 10', and you cannot make that choice without knowing what the numbers are. Technically you could choose between 'new and old' without knowing what the numbers are (as your DM is apparently asking you to do), but no other game feature asks you to make an uninformed 'choice', asks you to select between two options without knowing what the options actually are. There certainly could be unknown or unintended consequences of your choice, but you as a player know the mechanical effects of your PC's game features and what will happen 'on your end' when you use them. That's just how the rules work1.
Do they think that you don't know any hp totals? Or just temp hp?
Generally people play 5e with players having access to all their character's stats and rolls, including hp totals. More rarely, some DMs require that all rolls be made in secret (either they make the rolls themselves, or players make their rolls without knowing the result). This is the DM's prerogative and a perfectly fine style of play. What becomes suspicious is when some rolls are made in secret and others are not. The DM calling for Stealth rolls to be made in secret might be reasonable, since the PC would not know the result. But it would be questionable to have open hp rolls but secret temporary hp rolls. Let's compare the definitions. For hp:
Hit points represent a combination of physical and mental durability, the will to live, and luck. Creatures with more hit points are more difficult to kill. Those with fewer hit points are more fragile.
Temporary hit points:
Temporary hit points aren't actual hit points; they are a buffer against damage, a pool of hit points that protect you from injury.
Both of these are pretty abstract, and a game in which the players don't get to know either is a fair playstyle. It is harder to justify why you would know one and not the other, though. If your DM is saying that your "character is unaware of how effective the spell cast was", are they making this same ruling when the cleric casts cure wounds, heroism, or aid? When your party spends hit dice to recover hp on a short rest, do they get to know the result of their roll before deciding whether to spend another die and roll again? If all hp are unknown, fair enough - but if this ruling affects only castings of false life, that seems pretty targeted at just your PC. It is still the DM's call, but they should at least be able to justify why they have a ruling that specifically targets just your character and their class ability.
Do they think that this ability, or warlocks in general, are OP?
At-will casting of spells can, at first blush, seem like something that would mess with power levels, and if your DM has not run a game with a warlock before, this might be their concern. The first time I DMd a warlock and they selected at-will casting of mage armor, my first thought was, "The mage has to use three first level slots to have 24 hour access to mage armor, but the warlock gets it for free? That seems unbalanced..." As the game progressed, I realized how underpowered warlocks are in general compared to other classes (hexblades and 'coffeelocks' excepted). At 10th level, when that wizard has to use three slots to keep mage armor up all day, it is three of the least powerful of their fifteen slots - while the warlock is still getting only two spell slots (and recovering two more on a short rest while the wizard recovers five more). Warlocks don't get many spell slots; rather, they get invocations, and you have decided to invest in an invocation that gives you at-will casting of false life - that decision has been baked into the balance of the game. It's also worth pointing out that the few spells your warlock casts with their actual slots will scale with level, but regardless of their level, the false life will always be cast at first level. If the 3hp difference between 5 and 8 temporary hp seems important to your DM now, in a level or two it just won't matter that much. During a fight, when getting new temp hp actually matters, it still costs you your action to cast the spell, and so has a high opportunity cost. Between fights, allowing your warlock to 'cast until 8' is no more unbalancing than allowing the PC with the guidance cantrip to buff their Perception to listen at the door, and then using the 'at-will', 'no cost' spell again to buff their Initiative for the ensuing combat. That is, it is not unbalanced at all. If your DM is not allowing you to know your temp hp total specifically out of power concerns, you can certainly bring up these points.
Do they think that at-will casting breaks immersion?
Maybe your DM is trying to create an atmosphere where 'magic is special', and casting a spell is a powerful act. They worry that your at-will casting ('I keep casting until I hit 8') will somehow cheapen the experience of magic in the game. In this case, you can work with them to make sure your warlock is actually contributing to, rather than detracting from, the mood they are trying to create. Remind them that this is an invocation, not a spell, for you - part of the reflexive habits of your quirky or perhaps creepy warlock. Unlike priests, who generally have faith in their god, a warlock has a much more transactional or contractual relationship with their patron, and may not trust them at all. This lack of trust can make them superstitious or even paranoid about the supernatural forces around them (much like a a Vistani who is perpetually making signs to ward off the evil eye). Constantly using your at-will invocations, invoking the favor of your patron as a defense mechanism in a world where 'real magic' is powerful, could add to the atmosphere your DM is trying to create2.
Work with your DM
Respectfully ask your DM why they have made this ruling, state some relevant counter-arguments, and accept their decision. Good luck!
1As a long aside, several answers here also explain this in terms of the 'combining magical effects' rule and state that the 'most potent casting' of false life will be in effect, but I don't think that this applies. The combining magical effects rule is the general case for when you have two spells with overlapping duration, including two false life spells. However, the more specific rule is that of temporary hp, since that is what this specific spell does. As the passage cited above tells us, If you have temporary hit points and receive more of them, you decide whether to keep the ones you have or to gain the new ones. This is a specific-over-general exception to the combining effects rule and in this specific case you actively select which casting takes precedence. Further, you can keep the old temp hp or not - but if you are 'not keeping' them, you must be losing them, so if you chose to gain the new temp hp but the old casting was in effect, you would then have no temp hp. Similarly, if you chose to keep the old hp, you are choosing to 'not gain' the new ones. If the new casting was in effect you would also have not have temp hp.
As of the time of writing, those answers that have argued that the combining effects rule applies have cited only the first part of it when saying that the 'most potent' casting would take effect. None have included the full rule which also says
For example, if two clerics cast bless on the same target, that character gains the spell’s benefit only once; he or she doesn’t get to roll two bonus dice.
With the example included, it becomes clear that 'most potent' considers only the spells' potential effects before dice are rolled. Just as both bless spells have a potency of '+d4']2, both castings of false life would have a potency of '+d4+4': neither would be more potent than the other and you don't judge potency on the actual value rolled. Since both castings are of equal potency, if this was the rule used (rather than the more specific 'you choose'), you would always have to take the result of the new casting (since it is the more recent), regardless of what was rolled on either casting.
2 I don't consider myself superstitious, or a Christian - and yet I always say 'salud' when people sneeze, always cross myself when entering the aisle in front of the altar, and always bow to the shomen when stepping on or off the tatami - I imagine this might look odd to an outside observer, and a warlock could turn such minor rituals up to 11.