What are some methods that tournaments use to limit draws? There has been a rising faction of chess players who believe that professional chess is too drawish. What are some ways that tournaments fight against the prevalence of draws?
The problem with chess having a large number of draws cannot itself be solved - it is simply a result of the defensive possibilities in many systems and the fact that high-level play reduces the number of losses due to simple mistakes. Without a fundamental change to the rules of chess this is not likely to change.

There have been several measures to prevent top players from agreeing to draws before the positions are completely played out, which I'll mention below. One thing to keep in mind, however, is that draws by agreement are often used by players in long tournaments as a way of preventing fatigue or simply playing on in dead positions until someone makes a mistake, and it's up for debate whether forcing/encouraging play in such conditions makes for chess that is more interesting or of better quality, or if it detracts from the events in which those rules are used.

The most commonly talked about variant is what are known as 'Sofia rules' (introduced in a tournament at Sofia in 2005), which states that no draws can be agreed before the position itself becomes a theoretical draw, though this assumes that the arbiter is theoretically competent enough to realise that the drawn position is a draw, and doesn't stop the players deliberately trading down into such a position.

The main other option that has been suggested is to provide more points for a win than for a draw, thereby hopefully making the players want to push for a win as a win and a loss is still worth more points than two wins. There has been some discussion that this is unfair to players in hard-fought draws, however.

I'll also link you to this wikipedia article, which lists some other things that have been suggested over the years.