Do I need to watch the Hannibal movies before I watch the show?

The TV show depicts the events before the movie trilogy, in particular before Red Dragon (or Michael Mann's 1986 take on it, Manhunter, for that matter), showing Will Graham's work as FBI profiler under assistance (and care, given that the TV show depicts a significantly more deranged version of Will Graham) of Dr. Hannibal Lecter and thus the events before Hannibal's true nature is revealed and before his arrest.

Though, having seen not much more than the first few episodes of the TV show, I don't know if the later episodes might even venture as far as into the story of Red Dragon (or any of the later set/previous movies with Clarice Starling), but even then you don't have to watch the movies beforehand, it might even benefit the suspense if you don't.

The TV show is thus more or less a prequel to the movies (ignoring the rather infamous Hannibal Rising, which isn't necessary for the story either, though, being in turn set before the TV show). So while it may add to the enjoyment of the TV show to be acquainted with the characters (and see/compare the different takes on them), it is absolutely not required to understand the characters or the plot in any way. So no, you don't need to watch the movies, they are more or less independent (or rather later in the chronology).

Do you need to watch the movies? No. The show is self-contained and has its own variations upon Thomas Harris, let alone the movies. But will it enhance your ability to appreciate what's being done in the series by being familiar with the films and the books? Possibly.

The number of direct quotes from the books Red Dragon, Silence of the Lambs, and Hannibal are numerous enough that Bryan Fuller was awarding prizes to those who could list them at a NerdHQ presentation. And he had a lot of prizes.

There is also the fact that Fuller began the project with a seven-year plan: Season 1-3 were a prequel to Red Dragon, Season 4 was to be Red Dragon (looks like we're getting into that a little earlier with Season 3), and Season 5 would (if they could wrest the rights back) be Silence of the Lambs-era, and Seasons 6 and 7 would be the Hannibal-era. In addition, a number of events that have occurred in the series relate directly to what's to come. In fact, the opening crime scene of the pilot episode, "Aperitif", was meant to be one of the Frances Dolarhyde (Red Dragon) killings. And the characters of Franklyn Froideveaux and Tobias Budge were initially planned to be Benjamin Raspail and Jame Gumb, but had to be renamed/redone in light of the fact that they do not have the rights to Silence of the Lambs.

Another playing with expectations of the audience that is thoroughly instilled with Thomas Harris references is a flaming wheelchair--an event that is very similar to, yet departs from how it is done in the books/films of Red Dragon. And then there are the characters that Harris introduces and then drops from book to book being used again--but with unknown fates (e.g., Alan/Alanna Bloom). And, of course, the introduction of established characters like Mason Verger and Lady Murasaki have an added frisson of back history.

So, if you want to have more fun with the game, so to speak, becoming familiar with the other incarnations of Hannibal may be helpful. But, no, it's not necessary, any more than it's necessary to recognize that visual quote from The Shining that Fuller too such joy in concocting for us in the pilot to follow the action.