I have a 1080p video with 300dpi psd image that is resized on the timeline to be in the lower right corner. Looks great. Exporting to mpeg-2 dvd achitect default template, the image looks horrible. The text in the image looks bad.
A DVD is standard definition which is far less quality than your HD source so, unfortunately, this is not surprising.
Are you doing the resize using Track Motion or Pan/Crop (IMO, the preferred method)?
For starts, DPI is a print term and does not apply to video and the term should not be used when discussing video, it just confuses things. What are the dimensions of the graphic, in PIXELS? Your video frame is 1920x1080, for a point of reference. Rather than downscaling the graphic in Vegas, create a version of the image at the "proper size" for the video project. What that means is create a blank page in Photoshop or other graphics program at the video size of 1920x1080, and scale and position the graphic in the lower corner, as you did in Vegas. Now save that graphic page at 1920x1080, using a format that preserves the alpha channel (transparent background) such as PNG. When you import that into Vegas and place it in the timeline, it should appear as an overlay over the video background, no scaling needed.
But as Mike mentioned, you are then taking that 1920x1080 high-resolution video and scaling it down to 720x480 for DVD, so you're losing a lot of quality. If there is small text or fine details in the logo, that can be a problem due to the limited resolution of DVD. Perhaps you could put a semi-transparent black box behind the logo to help it stand out better from the background video? Another thing, turn on the Title Safe Markers in Vegas and check that the logo is not too close to the edge, or it might get cropped on some displays. May not matter for web, but does for DVD/TV viewing.
As pointed out above, a lot of this is simply due to scaling. Bad scaling will wreck the image, as will a bad upscaling back to HD on playback. How does the graphic look when played back at 1:1 on your monitor?
Another thing to double check is if the graphic is being compromised by interlacing in the encode or playback.