My DM told me when I hit a guy with 2 unarmed strikes (punches) that I have disadvantage on one of my hands. Is this true? Does a monk have a weaker off hand in D&D 5e?
No. There is no handed-ness in D&D, so characters do not have a dominant and weak hand. (Unless the player narrates them as such)
In addition, even when wielding two weapons, you don't have disadvantage on the second one you attack with (referred to as the off-hand, but either hand can be your off-hand depending on which weapon you swing first), rather you don't get to add your Strength or Dexterity to the damage roll.
And even that doesn't apply to Monks, who don't have to use both hands to make 2 unarmed attacks; their Flurry and Martial Arts ability allow them to make extra attacks, which they can make with their dominant hand. Or their face, if they prefer.
So no, there is no such rule. Your DM is making that up. Just keep in mind that the game does allow them to do that, but it's not a very nice thing to do this way. You might want to ask them whether they think this is an official rule (in which case they should be able to show you in the book somewhere) or whether this is a ruling they made. If it's a ruling they made, you might also want to ask them why they're adding this rule, as it (presumably?) makes the game less enjoyable for you if your Monk is suddenly much less competent.