My RCA TruFlat 27F520T 27" TV started acting up.

Check and replace damaged component/s at its main power regulator [SMPS] section circuit. Contact any service technician. If you wish to get some details; check the site linked here. Surf the site with patience. Pull up older posts.
http://electronicshelponline.blogspot.com/

It is probably the volume button.I clipped two prongs on mine and use the remote to turn it up and down

http://www.eserviceinfo.com/index.php?what=search2&searchstring=24F512T
http://www.devicemanuals.com/guide/Television/RCA/RCA-Tube-TV-27F520T-User's-Guide-TXpFeE16UT.html
Top one is a service and repair manual; the other is a user manual, both should assist.

You can take off the back of the tv. Look for the red wire at the top of the tube. Follow it down to the part that it goes into. On that part (transformer) there are 2 controlls. The one marked SCREEN, is the one you can turn up and watch the screen at the same time. If the tv shuts off, put that control back to the original position. If it brings up the brightness, then it is fixed. If not, take to a repair shop.

Buy a new remote.
Look around the tv and find the menu button.
Generally there is a separate power button.
You can also download the manual:
http://tv.manualsonline.com/manuals/mfg/rca/27f520t.html

i would honestly suggest if you still have the receipt return it back. dont forget your rating thanks.
This is a very common problem for RCA TV. I have one as well. or go to www.tv.rca.com/ there are free manual there

On the tv panel push and hold volume down until volume is all the way down and keep holding, then push the info button on the remote.

Unplug it for an hour and try again. This is an hard reset.

you can check only for the fuse if it is busted or not
locate the fuse. check if it open using a tester.......

if you have no experience yet on electronic servicing then you can fix it.

plug or socket loose or defective it might be due to power supply