Should an optical cable inside a house be run in conduit?

You want to use conduit ( flexible non metallic conduit) is the best for fiber use long sweeps for 90’s and you won’t crack your fiber. The conduit will also protect the fiber from damage during the instal. It is not recommended to staple fiber 1 over set staple or missed strike and the fiber is toast. For many years the only thing I used flexible non metallic was for fiber. Fled is about as easy but provides a safe path and put a pull string if you ever need you can pull a new cable or fiber at a later date.

Absolutely without question the industrial strength answer is install an empty plastic conduit pipe, and blow the fiber through later. Make all your bends very gentle so the pull is easy. The conduit needed is tiny and very flexible.

However, you may be using pre-terminated fiber, in which case the needed conduit starts to get large.

See https://www.youtube.com/watch?v=c8PgrvTrwAM for a guide to pre-terminated fiber. NEVER staple fiber. If you're framing, just drill holes and string the fiber from hole to hole. For your redundant backup fiber use different holes.

Assuming you're installing single mode fiber...

The other answers would have been correct many years ago, but developments in optical fiber technology have come a long way. Construction grade fiber is widely available now that is rugged and robust enough to be laid down like normal building wire. You can staple it, pull it tight around sharp 90-deg corners, and even wind it around pipe without any signal degradation.

Conduit is an option also, but it is absolutely not necessary any more - just make sure to buy the correct grade of ruggedized fiber for domestic construction applications and you can absolutely just run this like any other wire in your house.

You're more likely to break non-rugged fiber just trying to install it, to be honest, particularly if you're not experienced with fragile fiber - the rugged stuff you can't kill without a lot of dedicated neglect and abuse. It's made for builders with hard-toe boots, work gloves, power staplers, and thumbs bigger than their hammers.

If you're building a multimode network that's a different story altogether. I wouldn't invest in building multimode fiber into a building at this point. Go SMF - it will last forever. MMF - conduit, because you'll want to replace it at some point. At that, the extra you'll spend on the conduit and more-expensive multimode fiber would pay for the SFP+ SMF modules, so it seems silly to go cheap here. MMF is dead. I wouldn't bother investing in it at this point. It's not like you're a datacenter and have to shave every dollar off of thousands upon thousands of endpoints.

This isn't to say that you can't put plain SMF in a conduit, though - it does let you add more lines in the future, so that's always a plus, but as long as your SMF doesn't break you should never need to upgrade it - it will be good for 10GBit, 100GBit, 1000GBit, and whatever comes after that.