Will MN. IRS contact me if I've made a mistake on my state tax return?

You can check the MN refund site for information about your state refund.   If the state needs more information from you, or if they change your refund for some reason, they will send you a letter.

Federal and state refunds come from completely separate entities,  There is no rule as to which one will come first or how much time there will be between their arrivals.  Some states process returns quickly and some are very slow.

STATE RETURN

Make sure your state return was accepted:  https://turbotax.intuit.com/tax-tools/efile-status-lookup/

To track your state refund:    

ttps://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-refund/track-state-refund/L3jgO8PGs_US_en_US?uid=lt447ebr

https://ttlc.intuit.com/turbotax-support/en-us/help-article/state-taxes/contact-state-department-revenue/L9qVToi02_US_en_US?uid=m6e06um0