I own several rental properties. One I have owned for 25 years. I am about to buy a house in the same neighborhood to rehab and resale.  I was wondering if, tax wise, if it would be smarter to rent the house I am about to purchase after I do some rehab to it and then fix up my other rental and sell it so I can do a 1031 exchange. 

If you are going to flip the house, sell it outright. Too much trouble making it a rental then selling it on a 1031.

If you sell the house that you purchase and fix within a year, the gain is ordinary income. If you sell the rental and purchase another rental (or other like-kind property) with the proceeds, you will defer the entire gain. So, yes, it would be beneficial to sell the rental if your goal is to use the proceeds to buy another.

The buy and hold and the buy and rehab models are very different, especially when talking about tax computations.  It really depends on your own personal situation and tax circumstances.  Your thought process is sound.  Rehab the new one and then hold as a rental to demonstrate intent to hold for rental/investment purposes and then sell the current rental and 1031 Exchange into another rehab into rental, especially if you are planning on doing one property at a time, this would make a lot of sense.