I have the spell Find Familiar which partially reads:
You gain the service of a familiar, a spirit that takes an animal form you choose.
Your familiar acts independently of you, but it always obeys your commands.
This question suggests that your familiar will try to follow your commands to the best of its ability, but if the command is too complex for the simple mind of the animal to understand (DM decision based on their Intelligence) they can't follow through on it.
Would it be reasonable for a summoned Owl familiar to be on guard duty while my PC is asleep? Circling around the sleeping place or just sitting on a tree nearby and trying to wake up my PC when something out of the ordinary happens?
I could imagine this command to be worded like this:
Have a close look on our surroundings and if you see sudden movement or signs of other creatures like fire or sounds try to wake me up.
Things to consider:
The familiar obviously needs to sleep as well (as suggested by this question), this would happen the day after when my PC is awake by sending it to its pocket dimension for a reasonable time (8 hours probably).
The familiar is probably not able to determine what exactly is out of the ordinary so it might happen that it wakes the PC up for things that after a short investigation turn out to be nothing of big interest. Would these interruptions be too much for the rest to still be considered a long rest following the rules?
The familiar and the PC have been travelling together for a long time. The familiar is not really a wild animal anymore but dependent on the PC and has probably been trained to a certain degree like a pet. It will be rewarded for keeping guard and waking up the PC for positive reinforcement.
The owl has an intelligence of 2 which is not exactly to be considered overly intelligent. Comparing to real life this fits the fact that owls are not to be considered clever, even among birds. However there are trained owls that can at least do basic things like flying to certain places and returning in a given time-window.
As nwp pointed out the familiar is not actually an animal but a celestial/fey/fiend instead of a beast. Could this impact the intelligence? The spell description states the familiar has "the Statistics of the chosen form" which would include Intelligence.