"My dogs do not eat that."
Translation:To mí psi nežerou.
Well, look into current discussions and ... magic ... https://forum.duolingo.com/comment/30304525
Keep in mind that many words can either be used in the weak (second) position or in the strong position.
Also, by searching the web for Czech clitics you will find some scientific resources. They might be too advanced for you though and they use some linguistic lingo
https://ufal.mff.cuni.cz/~hana/bib/hana2008-clitics-sls.pdf (see the last slide)
For the positions see https://forum.duolingo.com/comment/30304525 but as I wrote above, it can also be in the strong first position if stressed.
With "To nežerou moji psi" depending on the pronunciation you could be suggesting that YOUR dogs do not est that (unlike some other dogs) or that YOUR DOGS are the subjects that do not eat that (unlike someone else). It is not usable for the information whether your dogs do or do not eat that or whether or eat that or some other thing. I will add it.