In the Monster Manual it does not say you can't, but it does not say you can either. Сan player characters be turned into zombies? And if they can, what changes?
Yes, but maybe not like you think
In Dungeons & Dragons, zombies are not contagious and being bitten by one does not turn you into a zombie. This is why you do not see anything in the Monster Manual about players being turned into zombies.
Zombies are created primarily through use of the Animate Dead spell. This spell targets "a pile of bones or a corpse of a Medium or Small Humanoid within range." Players qualify, so can be turned into Zombies by doing this. What changes is basically that the player character stops being a player character and becomes an NPC monster under the control of whomever cast the spell.
(Of course, in order to be targeted by this spell, the player character already needs to be dead.)
The steps are:
Cast Animate Dead on the corpse.
You now have a zombie under your control and the player gets to make a new character.
D&D Zombies are not TV Zombies
D&D zombies are corpses animated by magic - they are not animated by zombie viruses and are therefore not contagious.
Here's a bit of background that might be helpful. D&D was initially written and published (1974) just a few years after the release of the movie Night of the Living Dead (1968). As Wikipedia notes:
Although the film is regarded as one of the launching pads for the modern zombie movie, the screenplay itself never uses the word.
What is used, in a news report near the end of the film, is the word "ghoul" in reference to the monsters. And that's reflected in original D&D: ghouls may turn victims into more ghouls (after killing them), but zombies don't do that.
Per the Monster Manual: a Wight can when using life drain
When the wight uses the life drain attack, it may end up killing the character. If the character is a humanoid then the character slain by life drain can be turned into a zombie by the wight.
A humanoid slain by this attack rises 24 hours later as a zombie under the wight’s control, unless the humanoid is restored to life or its body is destroyed. The wight can have no more than twelve zombies under its control at one time. (MM, Wight; Basic Rules, p. 159)
I have done this via a wight as the DM twice in the past few years; once to a PC and once to an NPC. In both cases the party encountered them during a subsequent adventure. (As an aside: a wight leading a band of zombies is a fun encounter, or can be).
A simpler way is to use the animate dead spell on the PC's corpse, as noted in @Erik's answer.
Normally, their stat block is now identical to the zombie stat block, and they become an NPC/Monster under the control of the DM.
If your players want to play as that monster, they can if you let them but there's no mechanical tool available for them to have classes, get levels, etc. In that respect, do what's fun. (I mean, how much fun is it to work for a wight? Depends on the player, I guess).