Yes, I had this exact same problem with mine after I had it installed at the Ford dealer. My truck is a '06 as well and I had the Ford factory Gold system(Alarm/Remote Start) installed at the local ford dealer. Problem is that the tach signal (which is what the remote start utilizes to sense that the truck is running) is so weak on these trucks, you need what Ford calls a tach signal booster to bump up the tach signal voltage. Not sure of the exact part number, but it is a Ford part. Here is the technical service bulletin that Ford put out to address this issue...
Intermittent Unsuccessful Remote Start and/or Stall After Remote Start:
Some trucks may exhibit a no start condition when using a remote start system. Or, the the engine may start and run for up to five seconds then stall. The start/stall condition may repeat itself up to four times without operator input. The technician should contact the Ford Remote Start Technical Hotline for diagnostic assistance. A tach signal booster, P/N 102818-2, may be needed. Broadcast Message 1722.