According to Apple's specifications, you are required to have an HDMI cable and a TV with an HDMI port. The source video coming from the AppleTV is digital. Thus you could connect via HDMI, or a cable that converts HDMI to DVI since both are digital signals (Maybe check and see if your TV has a DVI port instead). There is a small possibility that the Apple TV is also outputting an analog video signal but I highly doubt it. The only way to know for sure would be to try it with a VGA to HDMI cable. I would bet you would need a powered converter that converts a digital video signal to an analog video signal and that a simple cable will not work. Also keep in mind you will need a separate optical audio cable as VGA/DVI does not transfer audio. If your TV does not accept optical audio then you will need a receiver that does and a set of speakers. You might end up putting more work/money into making it work with an older TV than simply purchasing a newer TV.
Welcome to the Apple Community.
Some users with DVI have managed to get their TV's to work with DVI-HDMI cables. DVI carries no audio, so alternative connections need to be explored to enable audio. DVI doesn't necessarily support HDCP as well as other standards used by HDMI (which may or may not be an issue)
There are hardware converters that will convert HDMI to various other types of output, however there are some issues with doing so that you should be aware of.
Resolution and aspect ratio
I'm not aware of any converters that will scale the output from the Apple TV, any TV or projector which is used will need to be widescreen and support resolutions of 720p (Apple TV 2), 720p/1080p (Apple TV 3)
Sorry that I am posting this so late, but I stumbled across this today and remembered you needed a solution for this. Check this out: