Obviously they need maintenance to remain habitable. But do skyscrapers require maintenance to remain standing, within the expected lifetime of their structural materials? What kind of maintenance? What would tend to fail first (absent earthquakes)?
With no maintenance at all I’d guess (based on the series Life Without People, anyway) that they wouldn’t last more than 100 years, if that. You can look at buildings in Pripyat to see the progression, but basically the windows break, water gets in, the concrete goes through expansion and compression with the seasons and as the water gets into it it cracks and also the reinforcing steel rots and expands, the eventually you have a collapse. If you are doing maintenance, then you are fixing windows, painting, sealing and repairing the concrete to ensure it’s protected and stays strong, etc. Even in those conditions I’m unsure how long a modern sky scraper could last wrt to the OP…I doubt for 2000 years, though I guess if you took extraordinary care it could happen, especially if they develop things like self repairing concrete or some sort of nano-bot technology that could penetrate the concrete to repair or reinforce the substructure. Even without that you probably could keep, oh, say the Empire State Building up for that time if you really, really wanted too, though the cost would be huge I imagine and you’d almost certainly need to do a full renovation every century or so. You’d end up in a few thousand years with the equivalent of Thucydides ship (or, I guess, Thucydides Empire State Building), sort of along the lines of Newgrange, in that the original structure has been changed to suit modern tourists to give the impression of what once was. I think that if our civilization continues relatively unbroken for 2000 years you would see a few, select buildings or structures that get this treatment, renovated every century or so, and geared towards tourists would would come to admire this thing from a past age. It would be interesting for us, from this time to see them…I’d guess we might not even recognize the buildings anymore, except in the most general structure, and probably laugh at what the tour guides would say about them. 
I’d think pretty much everything that falls into the category “Giant holes in the ground” will survive. How impressive they will be will depend on how fancy the hole is.
I toured the initial tunnels for the Yucca Mountain nuclear waste repository just before its funding got cut, and I’m pretty sure it will still be there in a thousand years, but it’s not really visually impressive.
The Cheyenne Mountain Complex will probably be more popular