IT STOPPED PLAYING BACK. it worked fine but all of a sudden, when ytou playback the recording, a blue screen appears and no picture or audio is heard.

have you connected up correctly, have you selected correct EXTERNAL inputs, you do NOT "tune" a Channel to see picture, you choose external AV input on set. Must connect color to color. 3 for Composite Video AV.Yellow, Video, Red & White Audio. and 5 cables for Component 3 for Video , & 2 for Audio Blue, Green, Red, Also Red & White, Audio.
AV

If your Sony DCR-HC14E shows fine picture in playback of old tapes and does not give live view in camera mode to record then its CDD Imager problem that Sony will repair free of cost.
Click Sony CCD Imager Support for details.

Sorry to say- but it is trashed.. It would cost more to fix it than it would be to buy a new one.

One possible cause is that normal recording
could not be performed because the heads were
dirty at the time of recording. Clean the heads,
and perform recording and playback again. If
normal playback can be performed, the heads are
clean. Before an important recording, be sure to
conduct a test recording to make sure that normal
recording can be performed.

≥If the heads become dirty soon after cleaning,
the problem may be the tape. If so, try another
cassette.

≥During playback, the picture or sound may be
interrupted momentarily, but this is not a
malfunction of the Camcorder. (One possible
cause is that the playback is interrupted by dirt
or dust that momentarily adheres to the heads.)

tape path is out of alignement

You can see an image in a camcorder viewfinder any time the camera's actually on in record mode, whether or not the camcorder is actually recording. So that's no guarantee.

Assuming the obvious (you have the right kind of tapes, they're not in write-protect mode, you rewind before playing, etc). I recommend that you borrow a tape of the same kind from someone else and check it on playback in your camcorder. This should lead you know understand if you have a playback problem or a recording problem.

I had the same problem with the DCR-TRV250 where it just stopped playing back tapes. It appeared to be recording correctly but upon playing it back, all I got was the blue screen. I tested the USB, SVideo, and DV outputs and got the same blue screen.

After weeks of deliberating, I finally decided to try cleaning the heads and that cured it! I removed the screws that allowed me to remove the transport assembly cover, which made it easier to reach the heads with a Qtip. I cleaned them using rubbing alcohol and everything seems fine now.

The camcorder may be in what may be called "narration mode." This function mutes the playback audio so you can narrate with you own voice when playing tapes back. Check for a function like this in the playback sound menu.

a sony cleaning cassette may clear the gunk from your friend ' s possibly ? bad tapes ( do not rewind it - or the gunk will be redeposited ) ; if still no video , get an estimate to have your heads and guides cleaned . . . professionally ;

I covered several threads that recommended buying a Mini DV cleaner and running it up to 4 times. They also suggested to change to change your video tape as it may have been the cause of the original problem.

These solutions may have worked for many but not me. My camera was becoming progressively worse. It went to blue bars and no sound to no picture at all...just blue. After using a cleaner 4 times I became frustrated to the point I was going to buy a new camera.

However, with the camera still playing, I did the typical guy thing and rolled up my fist and gave it a good hit along side of it. Remarkably it went back to just the blue bars. I hit it again and it went back to a perfect picture. I'm not saying this will work for you and hitting it too hard could make things worse but I was at a point where I didn't care any more and took a chance that paid off.