Not really, I don't think the sweat would damage your hair too much, as long as you're not manipulating your hair too much or try to touch it up with a curling iron, for example. You should just put a little moisturizer on your hair everday and/or a shine spray, and it should look pretty decent by the end of the week.
It won't damage ur hair but depending on the amount of new growth that u have, the strawset will not last longer then a week, ESP. if u sweat alot during work outs. If will puff out at the roots, and won't look as 'neat' as it should.