I have a Pioneer CLD-D703 which plays CDs just fine, however it will not play LDs. When you load a LD, it just groans and never spins the disc. Then the tray re-opens. Here is a video of the issue:
http://youtu.be/0kk37IzkrOY

Any suggestions?

That sounds an awful lot like a load motor under stress. It sounds as though there's not enough belt tension to raise the laser assembly and close the clamping system, even though the tray will still open and close. As I recall, the same motor, belt and pulley system does all that on the CLD-D703, though I don't have one right here to check that out. A CD would be so light that it wouldn't cause the problem to arise. When you turn the player on and off without a disc in the tray, you don't have any trouble with the laser assembly moving into place, do you?

I had a similar problem with a CLD-D515 and initially I was able to manually assist the raising of the assembly with a tug upward on it at the moment of the "groaning" noise in order to get the disc clamped in place and spinning. I suspect this is the same problem you are having. The proper solution however was to replace the load belt with a good one, and clean the two pulleys so there was no slip and a good grip. Also relube gears and tray where necessary.

Normally if the tray reopens the unit never finishes the loading cycle which would be a cleaning the loading belt task. If the load cycle completes and you hear a groan then it is must likely the spindle motor but could be the power supply.

Take off the top and then load a LD. If the noise sounds as the mechanical assy is rising then grab it and help assist it up and see if it will play. If the load completes and you hear the pickup move and then the groan then you are looking at either the spindle motor or power supply.