I'm not sure if membranes can dry out and/or if they can be rehydrated and/or if there would be any damage from being dry, but you could certainly try leaving it on for a while (and maybe power cycle it a few times so it runs a bunch of water past the membrane) and see if it manages to push through the membrane.
As for the water coming back out, I wouldn't worry about that until you figure out why it's not getting past the membrane section. If the water can't make it to the end, it's going to build up a bit of pressure, especially if there's still air in the lines, and it's going to have to go somewhere.
Just make sure you double check all your plumbing. If you're sure nothing has changed since the last time it worked, that's great, otherwise make sure to check over all those lines. I tried, but I think one of those lines changes color so it's hard to tell. For example, the far left carbon filter has a yellow line coming out of it. It should be connected to the top right port on the pump, where I see a yellow line.
Something you could do is pull the line out of the membrane inlet and run the pump with that going to the sink. Make sure the water is making it that far. If it is, then pull the two lines waste lines, one at a time, out of the pump and they should both have water coming out of them. Find out where the water is stopping (in the pump? the filter? the tube?) and work backwards until you find the problem or at least find out where it is and we can go from there.
