re: Will the summer heat eventually kill out dollar weed?

Dollarweed is a perennial weed - it won’t die out on its own. You are going to have to re-treat to get it under control. Also make sure your cutting your lawn at the right height, and if you have low spots, a poorly draining lawn you’ll need to fix that over time.

If you are persistent you can get it under control, but it will take time, think months/year, not days/weeks.

Here is a nice article with all your control options.

P.S. make sure it’s dollarweed and not ponyfoot (dichronda) - sometimes people confuse them.
Dollarweed Control