My 19 inch Apex tv will not turn on with of without the remote Should I just throw it away or is there something I can do to fix it?
First, I would suggest listing it on Craigslist or your local Freecycle if it's determined that you can't fix it. Generally, someone out there can at least scrap it out for parts or recyclable materials.
Now, to the original question. Was it working prior to hooking it to the VCR to begin with? Or did it stop working before you tried hooking it to the VCR?
There are multiple things that could be wrong, the worst one being that something burned out inside the power supply in the TV. Generally, if that happens, the replacement part is higher than buying a new TV, unfortunately. You could search you local yellow pages (physically or online) to find a repair shop, and call them for advice. They will often tell you whether or not something is worth fixing or not, or at least give you a price estimate, and you could decide whether it's worth it.
If it's ruined, I would definitely suggest listing it on Craigslist or Freecycle, and leave it out on the curb or in your parking lot, where whoever wants it need never meet you in person.