Is TurboTax supposed to update your refund amount from the unemployment tax exemption automatically?

The IRS, not TurboTax will recalculate your tax return for the unemployment exclusion and send you the additional refund. They will do it automatically if you received unemployment income, but you did not have the exclusion on the tax return that you filed, and your Adjusted Gross Income (AGI) is less than $150,000. You do not have to do anything to get the additional refund. It might take several months to get it. For details see the following IRS announcements.

IRS to recalculate taxes on unemployment benefits; refunds to start in May

IRS begins correcting tax returns for unemployment compensation income exclusion

If the tax return that you filed did not have the unemployment exclusion, as of now I believe that TurboTax is leaving the tax return in TurboTax Online as it was when you originally filed it. So you will not see the updated refund amount in your tax return in TurboTax Online.