My property is rented by a management company. I receive a portion of the rent. I continue to make mortgage payments. Do I submit both a 1099-MISC and a Schedule E?

You enter the rental income you receive only once.  If you enter the 1099-MISC using the 1099-MISC interview TurboTax will notice that you used Box 3 for the dollars and ask you if the amount should flow to Schedule E.  Or, you can simply enter the amount over in the "Rental Properties and Royalties" interview. 

(Just so you know... There's absolutely no need to enter 1099-MISC information actually using any of TurboTax's interviews that ask for 1099-MISC information.  What's important is that you report your total income correctly as no detail about individual 1099-MISCs is included in your income tax return.)

If the management company sends you a 1099-misc then use that information for the Sch E ... you do not need to use the 1099 input screens.