The first thing that I should mention is that I don't invest turnkey out of state. However, I do invest out of state, and wanted to mention a few things that I've found helpful to avoid biggish mistakes.
(1) Always look at the numbers. Even after a purchase, you should be getting statements on a monthly (ideally) not less than quarterly basis. Review the reports when they come out, and question anything that looks out of place.
(2) I like to be more hands-on in bigger capital projects. I've flown into the market that I invest in to be actively involved in the bidding process for capital projects. Otherwise, you might not get the best deal, or the best contractor for the job. On one job in particular, I met the contractors and handled the bidding process directly. It also made quite an impression with the tenants, as they knew that I was out of state and made a trip in specifically to deal with the major issue.
(3) Ask for pictures of repairs being done, especially early on in the relationship. It provides a level of accountability, and provides a bit of security to ensure you're paying for repairs that were done. Otherwise, tenants might think you aren't addressing any maintenance issues and can lead to higher turnover.