My TV picture has a greenish tint.

The first thing to check is the cables connected at the back of the TV.

Check cable connection, and try using a different cable, video signal is often coded in separate colors signal, and a bad cable can cause a green picture when red or magenta (depending on type of signal) are out.

A faulty receiver or DVD player can also send a bad output. Try connecting different devices to TV.

If the cables and receiver are OK there are also internal faults that can cause the same effect.
The fault depends mainly on the model and type of the TV set.

On all TVs a faulty input/output can cause greenish picture.

On all digital TVs the signal is also elaborated on different color channels . Video circuits are often located on digital board.
A fault on digital video processing can cause a green picture.

If you have a rear projection CRT based technology, one of the three CRT may be faulty, or the power board supplying the CRT (red in your case) may have blown.
In that case you will get a greenish picture.

If instead you have a CRT TV, a greenish picture means that the red gun inside the cathode ray tube is not working. The CRT itself must be replaced.

Generally, when the problem is not the video cable , or the connected device, the TV must be inspected and repaired by a technician.

If you have a new digital TV, as an LCD, Plasma, or any non CRT based technology, the problem is usually caused by cables, input output, or connected devices.

Check the cables thoroughly, if that is not the problem, call manufacturer to find the nearest repair service location.