The word "observation" may be misleading you here. What matters is whether there is any interaction of any kind (for example, the particle collides with a random air molecule floating around somewhere between source and screen, or light shines on it, or there is a detector in one ofthe slits, or a polarizing filter behind one of the slits, or... ) whose result requires that the particle had to go through a particular slit. If there is, then the particle will behave as if only that one path was available to it. If there is not, then the particle will behave as if both paths are available to it.
The experiment in which sometimes both paths are possible and other times only one is possible has been done many times, and always matches the quantum mechanical prediction: An interference pattern is formed by those particles that could have gone through either slit and is not formed by those that could not. It's worth mentioning that you can't see this with a single particle, as each particle just makes a dot on the screen either way. To see the interference you have to run the experiment with a large number of particles and watch the individual dots build up the pattern.
If you google for "Kim delayed choice quantum eraser" you'll find some good descriptions of one of the more interesting experiments along these lines.
And if you are wondering why we keep on hearing about "observations"... It's historical. Physicists were using that word early in the 20th century when these concepts were first being thrashed out. By the time is became clear that "observation" was the wrong word it was too late - that usage had caught on in the popular press.