That's a tough one, but I'll try to explain it as well as I can.
Basically, DVDs suck. They utterly lack the freedom and flexibility that one has when using a media server setup or anything wholly digital.
Due to the fact that DVDs (and BluRays, for that matter) underly a strict set of rules when it comes to encoding and folder structures, it's easy to encounter issues when trying to manually include subtitles into a self-made video DVD.
Most importantly, .srt files are NOT supported anywhere on a DVD, since they are entirely text-based and hardly any DVD reader knows what the hell to do with that.
On every retail DVD/BluRay, subtitles are included in an .idx/.sub structure. Unlike .srt files, these are image-based.
That means, that there are two files (one small, one huge) with one containing images including the subtitles (already formatted, looks like a screenshot of some sorts) and the other one containing timings and the respective references to the fitting subtitle image.
So, as to what you can actually do to make it work for you:
If you have .idx/.sub files, it's rather easy. You can include them with almost every major DVD burning suite.
If you have .srt files, you're shit out of luck, since things get a little bit tricky from here on.
With .srt files, you pretty much have two options. Either you try to convert them into the .idx/.sub structure and add them more or less natively (that one's a mess), or you hard-encode (or 'burn') the subtitles into the video stream. Doing the latter, the subtitles become one with the video file, allowing them to appear everywhere the video file itself is playable.
There's two major downsides with this method though:
1.) You are NOT able to deactivate or get rid of the burned subtitles once they're hard-encoded, since they are permanently added to the video stream. Everybody who tells you otherwise is straight-up spreading nonsense and false hope. Sure, there are tools who claim to be able to get rid of burned subtitles, but all they do is put the video stream under surgery and forcefully cut out the parts where the subtitles are shown. That leaves you with a messed-up and weirdly cropped video file and a lot of frustration.
2.) If your video stream is low-res (240p/360p/480p) and you want to watch it on a big screen, things can get mushy.
When burning subtitles into a video stream, they obviously are bound to respect the video file's resolution. If you now try to watch it on a bigger screen, since the subtitles become a part of the video stream, they also get stretched to fit the display's size and resolution, causing them to appear out of focus and kind of ugly.
There you go. I hope I could shed some light unto your issue. If you choose to hard-encode the subtitles, I'd recommend you using Handbrake, since it's a free and extremely reliable tool and my personal favourite for basically anything video-related.
Edit: Okay, I just noticed that this thread is as old as the universe. Just do what oss recommended. I'll just keep this reply here solely for the extra bit of info included.