Is the landlord I rent from supported by taxpayer money because I’m a public employee that rents from him?
You can't have it both ways. If it's the employee's money, they have the right to do what they want with it, whether that means giving a portion to a union or keeping it for themselves.
If they don't have that legal right, then the money isn't theirs, and it's coming from taxpayers, because it's paid by the government.
If your rent is automatically withheld from your paycheck as terms of your employment and given directly to your landlord, then yes, that logic might hold.
However, I'm betting that's not the case.
1. Unions are not government funded. 2. Those who benefit from collective bargaining ought to help pay for it.