Of course, some movies deserve a remake given the advances in production budgets, technology and special effects. The nearly iconic Godzilla is an obvious example of that. War of the Worlds, Planet of the Apes and The Time Machine were good for the same reason. Psycho remakes were good but originals do not need to be discarded, (it's hard to beat the original Norman). Remake of Steinbeck's Of Mice and Men (1990's) was excellent and In the Heat of the Night was good. The Thomas Crown Affair, Ocean's Eleven and The Italian Job were excellent as originals and their respective remakes were also notable. Some "remakes" from old television series to movies have been top notch. e.g., Mission Impossible, The Adams Family, Get Smart, The Saint, My Favorite Martian, The Avengers to name a few. Some TV shows from the 60's sucked back then and subsequently suck when made to movies, visa-vi: The Beverly Hillbillies.
I would go see a remake of The Great Escape but who could fill Steve McQuein's shoes or ride a motorcycle like that? Maybe Russell Crow?
It depends on the remake...Some are good, but then it's annoying to hear people say things along the lines of "the only reason people like this movie is because of the special effects!! The originals are just better!"...And then there are people that get mad because this generation won't know about all the old movies...But I digress...
There honestly aren't really any movies where I have seen both the original and the remake...As far as just remakes I've seen...I liked War of the Worlds, Day the Earth Stood Still, Bloody Valentine 3D (that's a remake, right? I think so...), and some others...I haven't seen too many originals...Or any that I can think of at the moment...
Most remakes really don't work, especially as a lot of remakes are of good films. If it is a bad film people don't remember it to remake it. The original Italian Job is better than the remake, I certainly agree with that. Still very enjoyable though. To be honest I haven't seen many, but by all accounts there are a few that are better than the original but not many. The most quoted one for this I can think of is Ocean's Eleven. I have never heard anyone call it worse than the original.
Normally I have seen the remake but not the original. Most films where it is a remake that I have seen have been bland and dull compared to films that are new, and I am saying this without the original film to compare it to. A few that I really enjoyed stand out, like The Thomas Crown Affair and Casino Royale (if you can consider that a remake, and not simply the story told properly), but most aren't that good, even without a frame of reference.
In terms of good remakes look at the TV. Battlestar Galactica takes its old and roughly related former self and gives it a sound thrashing. I am still not sure if this is sci fi or a drama set in space. Thinking about it, the latter is more sci fi than most stuff called science fiction (when it should be called space opera).