Do I have to amend a return because of The American Rescue Plan that makes the first $10,200 of unemployment payments non-taxable? Or does the IRS do it automatically?

There is no need to amend, the IRS will recalculate the taxes based on the unemployment exclusion and send a refund if you are eligible.

Go to this IRS website for information - https://www.irs.gov/newsroom/irs-to-recalculate-taxes-on-unemployment-benefits-refunds-to-start-in-may

The $10,200 unemployment retroactive to 2020 was signed into law.

If you already filed - do not amend. See this:

https://www.irs.gov/newsroom/irs-to-recalculate-taxes-on-unemployment-benefits-refunds-to-start-in-may

You generally do not need to amend for Federal.

However, taxpayers would have to file an amended return if they did not originally claim the EITC or other credits, but now are eligible because the [unemployment comp] exclusion changed their income.  These taxpayers may want to review their state tax returns as well.

IRS to recalculate taxes on unemployment benefits; refunds to start in May | Internal Revenue Service

See these re State:

How is my state taxing unemployment income? (intuit.com)

Do I need to amend my state return to update my un... (intuit.com)