Recent research suggests that the Saharan desert formed around 7Ma.
Previous guesses for its age of formation were younger at 2-3 Ma. This means that tectonics plate movement is very unlikely to have formed the Saharan desert as it is only moving north at 2cm / year. This is roughly 20km per million years not quite enough for massive climate shifts.
The reason the Saharan exists is thought to be because there was once a sea that covered the Mediterranean and black seas called the Tethys sea. This is thought to have dried up around 7-11Ma.
This would have decreased the moisture available for precipitation during monsoons over the Saharan and over thousands of years dried the region up.
This is exacerbated by the Saharan's latitude which means that the Hadley cells that move air around the earth only bring dry cool air into the desert this does not carry moisture to the region. goo.gl/gv6z4z
The desert looks homogeneous because the main factor that controls geomorphology of the area is wind so you mostly get dunes, lunettes, etc like the link below. If water had a larger influence it would likely have plant matter growing and covered the surface so wind wouldn't be blowing sand all around. goo.gl/X3M1fk
Predicting the future is somewhat faux pas so I won't say what could form a forest where the desert is today just that there are plenty of moist regions at the same latitude so it is possible.
Finally I don't know who you are talking to but Mars most definitely does have dune systems and they are extensive.
http://www.nasa.gov/content/martian-sand-dunes-in-spring
The photos you see of mars are likely from curiosity rover and this rover was landed in a place with very interesting geomorphology so that they could study its history. They wouldn't learn much from landing in a dune field I fear.
They landed curiosity in an old meteorite impact crater right on the border between the northern lowlands and the southern highlands. The crater was filled with water when Mars had liquid water and this deposited the sedimentary layers you see in curiosity's pictures. These layers have since been eroded by wind.
For homework take these two links and find some dune structures on Mars by setting it to visible or maybe infrared.
goo.gl/SKx3ix
Happy hunting