Yes, it is possible. In fact, some people let their TVs handle the de-interlacing of DVDs rather than the DVD player, because it looks better (TV may have better de-interlacing capabilities than the actual DVD player).
That being said, if the DVD player is old, it's worth spending the money to get a decent new DVD player to compliment the new TV. There are even upconverting DVD players that will rescale DVDs to 720p or 1080i resolutions, and output the signal digitally to the TV (getting rid of the need for digital->analog->digital conversions). This doesn't make DVDs look like hi-def, per se, but it can make them look better, especially combined with the DVI/HDMI digital connection.
That being said, if the DVD player is old, it's worth spending the money to get a decent new DVD player to compliment the new TV. There are even upconverting DVD players that will rescale DVDs to 720p or 1080i resolutions, and output the signal digitally to the TV (getting rid of the need for digital->analog->digital conversions). This doesn't make DVDs look like hi-def, per se, but it can make them look better, especially combined with the DVI/HDMI digital connection.