If your monitor is going into stand-by as soon as the G-man starts talking, that's a software issue (as in drivers, bios, etc).
However, since yours is turning off after a certain amount of play, I would suspect heat.
There's also a number (a great number) of other things that could be wrong. For instance, when the G-man is speaking, it's a different video experience than when you step onto the train. One is a scripted scene with a pre-build video playing, the other is the physical world. You may be catching a hitch that way. Also, you could have a corrupt installation, throwing out a bad value to the computer when you get "released" into city-17, which can crach who knows how many things.
Another thing you can try (if possible) is another monitor. I know on some games you can sometimes run into problems with the monitor's capabilities and settings.
If worst comes to worst, you can try contacting (email or phone) your video card maker.