Will the summer heat eventually kill out dollar weed?

Yes, don't sweat it.

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

MSMA

Don’t know about heat knocking it back this summer as I spray mine when I see it, but I’d suggest you not give it any additional opportunity to spread. Pretty much any perennial weed is going to require multiple herbicide applications, often at 2 to 3 week intervals, to get it under control. But I’ll caution, make sure you read the label on any herbicide you use, also read university based fact sheets, on control of the weeds you are dealing with. Don’t hesitate to contact your parish or county university extension agent for further advice, they are happy and paid to help, and a great resource you pay for with your tax $.

Pre-emergent in fall, great for any number of fall annual weeds, but not perennials like dollarweed because dollarwwed propagates and expands by methods other than seed. Pre-emergents kills seedlings once the seed germinates.

Control of ponyfoot (dichronda) is similar to dollarweed. Here is some additional info on control measures from the LSU AgCenter which includes ponyfoot.
LINK

Sounds like a plan. Make sure you are using surfactant in your spray solution because dollarweed has a very waxy leaf.

Personally never tried Fertilome Dollar Weed Control as dollarweed never been a particularly large issue for me but I do like the ease of applying granular products when available vs mixing spray solutions.

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problem for the past couple years has been oxalis.

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Another one of perennial weeds that is difficult to eradicate. I have a few spots and I spray it to keep it in check, comes back each year, but I don’t allow it to spread.

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uptick in burweed

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Seems that’s true for many - worst I’ve seen it my yard this year. Hosts on local garden radio programs have said the same.