Just like the title says: how long does harvested yeast keep? I am assuming to keep the yeast you have to store it in the refrigerator, right?
In Yeast (White & Zainasheff), they have a table of the reliable and max shelf life of yeast storage techniques:
… &c., up to professional freezing.
People report success with fridge-stored slurry on much longer time frames. I think White & Zainasheff are being somewhat pedantic; the amount of mutation you'll see in a handful of months is probably both very real, but also not particularly noticable to a homebrewer's beer or palate, all things considered.
You may want to look into yeast washing (using acid to help purify the harvested yeast) if you want to pursue that path.
I've been washing yeast for years now and I've never had washed yeast be completely dead, even after 9 months, but that doesn't mean you should necessarily use it. If you do wash yeast, ALWAYS prepare a starter. This will let you know if the viability is good (if it propagates) and smell the starter before pitching. I have ruined two batches by pitching 'off-smelling' yeast washes. I am too cheap to leave the process alone; I have saved literally hundreds of dollars, but even one bad batch really pisses me off... Just remember, any flavors you smell in the starter are going to be in your beer; make sure you like them! Easy enough to run to the homebrew store and by some dry yeast or a smack pack! Good luck.