I've experienced this once with hops debris clogging up the works. I ended up very carefully venting pressure from the keg, and while holding the pressure relief valve disassembling the liquid-out post. Once the post is off, I quickly removed the liquid-out dip tube, cleared the clog with a sanitized tubing brush, and reassembled the keg.
Unfortunately, it clogged again within one pour.
I repeated the venting/disassembly processes again. This time, before reassembling everything, I took my quick-connect apart and removed the pin and spring that shuts off the flow of liquid through the quick-connect when it is removed from the keg. Then, I attached the liquid-out post on the keg without the poppet. One problem with this is that as soon as you connect it to the keg, the system is "on" all the way to the faucet. If you wanted to remove the beer tubing, you would need to vent all pressure in the keg, then open your faucet to drain all the beer back into the keg. Once the beer drained back into the keg, you will need to carefully disconnect the quick-connect, then reassemble the poppet in the beer-out post and add the pin/spring back into the quick-connect.
It seemed to me that the clog was in the tubing at first, then at the poppet and quick-connect the second time. Pours after removing all of the shut-off portions of the quick-connect and keg post were smooth, but full of debris. After a half dozen or so pours, they started to clear up.
