If the GPS receiver is a direct conversion type, then the biggest possible cause of interference is to the on board GPS. Not desirable. But if it's a superhet, then we have a potential problem. If it meets FCC (US) requirements on oscillator radiation, it could well have an Intermediate Frequency such that the local oscillator blocks a DME channel. If it meets EU standards, possibly not - but only possibly, depending upon which stanadrds the manufacturer has decided to declare it compliant to.
As far as cell phones are concerned, any LO radiation if the receiver uses high side injection could kill a DME channel. A complication arises in that the immunity of cabling on a nice new aircraft is quite likely to be different to that on a 10 or 20 year old aircraft where corrosion, movement etc have happened.
The problem is that as far as aircraft systems are concerned, you KNOW what frequencies are involved. Bringing in unknowns (because you DON'T know the frequency plan of every possible bit of equipment) is another matter. Add to that the fact that there isn't just the fundamental radiation to worry about, but the presence of intermodulation products when authorised transmissions are made from the aircraft, there is a VERY good reason for banning the use of these things.
If you wonder about my qualifications, 'radeng' is a reduction of 'radio engineer'.