Are there websites or programs that permit a simulation of the night sky in the past and the future on an ordinary computer?
(For the past, I would be content with objects visible to the naked eye.)
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The two main free destop programs that I know about, Stellarium and Celestia, do not include the proper motion of the stars when they move forward and backward in time. At least according to documentation that I've seen.
These programs claim to do it but I have no experience with them:
Microsoft Research's Worldwide Telescope (Free) could satisfy those requirements and then some. It's a virtual telescope with a solar system simulator and runs both as a Silverlight web app or as a desktop app.
It's extremely customizable so you can customize it to only show objects of a specific magnitude as viewed from a specific location and during a specific time.
If you have an iPhone/iPad, there's also StarWalk (commercial), which gives you a 'palm of your hand' experience for viewing the night sky from any time and location.
Certainly, there are many such programs. I personally use Starry Night.
Declaration of conflict of interest => In fact, I liked the program so much I went to work with the company that makes it.
The first program I used to do that, is currently available and still free, as most modern software, allows the use of updated catalogs and the use of different times to see the simulation. Stellarium IMHO has a better rendering, even when I prefer the controls on Cartes du Ciel.
KStars for KDE/Linux, available under the GNU General Public License (GPL)
XEphem, available for free for personal use and publicly funded research (see the web page for details).
You'll have to compile it yourself though but runs on (amongst others) Linux and OS X.
Google Sky for Android does "time travel." It's free.
I wonder about your use of "in the past and future." When a star is at an appreciable distance from us, while we can still tell if it's moving away from or towards us from redshift (radial velocity), we can no longer measure how it's moving relative to the background of start (tangential velocity).
I've tried rewinding stellarium as far as it will go (the year 99998 B.C.E.) and some stars have certainly moved in the rewind. Others however, haven retained their original coordinates. The reason for this is unclear, as a star which doesn't move in the sky would be a freak coincidence (heading directly towards or away from earth). Presumably, stellarium doesn't have the required data to reproduce the tangential component of these star's velocities.
And even if they DO have this data, there's no way to really calculate (with any reasonable certainty) all of the gravitational interactions that have occurred, or are going to occur, over the star's lifetime.
For this reason, I would be surprised if it were correct in the distant past or future.
Give WinStars a shot.
Here are the chief characteristics of the software: a database of 2,500,000 stars;
a catalogue of 10,000 nebulae, galaxies and star clusters;
a direction of observation which is easily controlled by a mouse and in real time;
a precise representation of the observable sky from a point on the earth’s surface on a given date;
a 3D interface to give more realism to celestial objects;
a calculation of notable astronomical phenomena visible from an observation point on the earth;
detailed information about each object;
a calculation of the positions of the principal satellites of Mars, Jupiter, Saturn, Uranus and Neptune, of comets and of asteroids;
an outline of the celestial equator, the ecliptic, and a grid of azimuthal and equatorial coordinates;
a 3D outline of the planetary orbits
drives a large range of telescopes
compatible with the seti@home BOINC version
Internet resources are also available: updated comets and asteroids' elements, querying of DSS (Digitized Sky Survey) servers to obtain a photograph of that portion of the sky being displayed by the program, a notice of the visibility of artificial satellites, etc.