There is a potential grammatical ambiguity. In fact, I misunderstood this slogan the first time I saw it.
Scott Adams, creator of the comic strip Dilbert, also noted the ambiguity on his blog. See bullet point #3 at http://blog.dilbert.com/post/143789982926/clinton-versus-trump-persuasion-scores.
First, the above answers correctly describe the most common understanding of "Love Trumps Hate". "Love" is the subject, "trumps" is the verb, and "hate" is the direct object.
However, I know people who use this slogan ironically. It also can be read (aloud, at least) as an imperative sentence. The subject is the understood "you". Then "(should) love" is the verb, "hate" is the direct object, and "Trump[']s" is an adjective modifying "hate".
In that reading, then, the sentence is similar to
"You should love the fact that Trump hates some people."
A purist would require an apostrophe to make "Trumps" a possessive noun. But it's not very unusual to drop that apostrophe. See the Wikipedia page at https://en.wikipedia.org/wiki/Apostrophe#Omission and further down the page under the "Criticism" heading.