I have a 2010 jeep liberty the gate light is on in the dashboard and rear gate wont open with remote or push button on rear door

Had the same problem with our 2012 liberty, however, this false reading on the dash stating the rear gate is open, would illuminate the interior lights and set off the car alarm when you locked the door with the remote. The problem is that water is entering the gate near the flip up glass. The jeep corporation has issued a TSB about this issue and the will re seal around the gate, the window, as well as replace the gate sensor which was getting wet and supplying you with false readings. I take it your vehicle is not under the 3/36 warranty so the latch assembly will cost about 100.00 to purchase and is easy to install. I have heard of some people drying the sensor out and having it work for them. The underlying cause is a water issue and chysler did not issue a recall for this issue, which is annoying, but they don't consider it safety related. The problem is that your alarm will go off for no reason, and if your interior lights are set to go on when you open the gate, this will subsequently drain the battery. Thanks jeep. Wish you good luck, in your case, the latch is just wet and malfunctioning in a different way. Pretty sure that is your issue.

The gate lock has issues, get it serviced.
if a DIY person? get the FSM and read that page. under body chapter.