Generally speaking, no. There are edge cases (newer televisions with HDR support etc) where the answer could be yes but for most displays and at resolutions supported by DVI the answer is probably no.
In fact, single link DVI and earlier versions of HDMI are exactly the same video signal. Obviously DVI doesn't carry audio, but you can convert between them with just a physical adapter the video signaling is identical. Dual Link DVI will require an active adapter or a DVI display, but up to 1920x1200 DVI and HDMI are effectively interchangeable.
Viper GTS