The December 2014 Temple Run 2 update included the "Bankrupt" objective, which is to run 2000m without any coins. This is a harder version of the earlier related ones (Stingy- 250m, Miser 500m, Allergic to Gold 1000m) and with the longer distance is much harder. There is a good chance with the longer distance of getting into Narrows, Mine Tunnels, Waterslides, or sections where the path has fallen away and it is too narrow to avoid the coins. What are some good methods for completing this objective?
For this objective, you need to be lucky and skilled. It's like the shorter versions where tipping and jumping can get you past coin sections without grabbing any, but harder. There are a few running sections which make this impossible:
1) The mine car sections generally you cannot avoid coins on the single-track sections, so you are doomed if you happen to get a mine car section in the first 2000m.
2) The narrow board sections are basically impossible to run through without getting coins.
3) A "gold coin" or "magnet" power up above a gap in a bridge means that you will end up collecting coins.
So if you get any of those, you'll pretty much have to try again. The parts where the path is narrow can also spell trouble, although sometimes you can jump to avoid coins there is the placement is favorable.
A few things to note now after completing this objective on multiple devices:
1) The water slide sections are great for this objective, as you can dive underwater for long stretches and avoid coins and obstacles there.
2) In the earlier versions, you could intentionally die if you were about to get some coins unavoidably, and then resurrect and be further down the paths and hopefully be in a place where it was again easier to avoid coins. But with the present system of Resurrection rewards, many of the Resurrection rewards (any gold or the magnet) ruin the "no coins" aspect of your run. But if you have some gems to spare, you can reroll the reward until you either get the shield power up (useful for avoiding coins) or the running boost (tricky to continue avoiding coins but at least possible.) It may take a fair number of rerolled rewards for this to be useful, so probably it doesn't make sense unless you are already close to 2000m and just hoping to get a little further without coins.
3) It is worth jumping to get the Shield power up when trying to complete this objective, as that can help you survive while trying to avoid coins. The Gem and Artifacts Chests you can jump for but don't help or hurt, and the Coin and Magnet ones are bad and need to be avoided. Boost doesn't directly ruin things, but it generally makes it harder.
4) Those stone arches are difficult because they block the view of the coins past them, making it hard to know which side to be on. With the shield power up active, you can jump through the arch and see where the coins aren't and aim to land there to avoid coins.
I found that it took many efforts but it was nice to have a good challenge and a new objective. One thing is that if you haven't already completed the Mega Runner objective of needing 1000 lifetime games, you may make some good progress on that with all of the short aborted games where you accidentally grabbed some coins!
Edit: Some people recommend pausing at corners and such to stop to get a better view of where the coins are, but I didn't find that necessary.