Bread

If you seal the jar (as in air-tight) then eventually the microbes in the starter will use up the oxygen and go dormant. If you're unlucky then the gases given off by the microbes will cause the pressure in the jar to increase and your jar explodes.

I should clarify though - I keep my starters in a non-air tight glass kilner jar (normal kilner with rubber seal removed) and an old plastic peanut butter tub (we buy it in 1kg tubs) with the top not completely on. I use these because they are what I have, and they will do the job of letting gases in/out and holding a starter with room to bubble up a bit.
The other things I ferment* I generally use a jar and muslin/tea towel.
You can, of course, keep things in air-tight containers and the world won't end but if it were me I'd burp them at least once a day (I do this when making sauerkraut). Just take the lid off and put it back on to let the pressure equalise, or just leave the lids a bit loose.

I expect a muslin would do no harm apart from maybe your starter might dry out on the top of a bit. Also when I feed my starters I give them a good stir to break up the bubbles (carbon dioxide) and encorporate air and therefore oxygen.

*Kombucha, vinegar, ginger beer, mead - so far. I've only started fermenting things fairly recently but a lot of the same principles apply as it's all microbial action.