Harm or harms

In this case, harm is being used as an infinitive, and is not being conjugated: ". . . don't [you] let stress harm your relationships." The subject of the sentence is "you" which is unvoiced, and in this particular case, the sentence is an injunction--you are being told what to do or not to do. A different sentence might help you:

"He drives across town to get cheaper gasoline."

The subject is "he" and the active verb is "to drive"--"to get" shows the purpose of his action. So an infinitive can be used to show the intention or the purpose of the active verb. In your sentence, "to do" is the active verb and is conjugated for the second person. "Harm" is an infinitive which shows the purpose of the active verb, "to do," and the purpose is not to allow stress to harm your relationships.

By the way, there is something else going on in your sentence which makes this a little more difficult to understand, and that is the use of "let." If you substitute "allow" for "let," you use the full infinitive form:

"Don't let stress harm your relationships."

But . . .

"Don't allow stress to harm your relationships."

I suppose there is some rule which explains why this is so, but i'll be damned if i know what it is.