If it involves money you might get screwed. A permission is unlikely and even if it is just for fun it might get shot down.

So I shouldn't even post a playthrough of the game?

No, you can do whatever you want. You just have to be aware that it is not safe. Though the worst that will happen is that youtube deletes it.

So I wouldn't get sued? What would happen to my youtube channel?

If you monetize you can get sued if they really want

Worst case that could happen, youtube deletes your account (which is also mentioned as action in their terms) and you get sued. While this is often no issue it might be that for some trademarks a current company buys the rights and want to do something with it. Then they start looking for users.
While I have a non monetarized channel I noticed that when Disney bought Lucasarts shortly after the Audio of my XWING Intro capture got muted due to copyright reasons.

Also what exactly might happen depends strongly on the country you live in. So it can't be generalized.

So how do Let's Players make money? I guess they obtain permission or play games that are "safe" in their monetization policy?

Firstly, monetizing is not the issue. Some people would have you believe that not monetizing makes your videos/channel safer. IT DOES NOT. Not monetizing has NO EFFECT on the safety of your video material compared to monetizing.

Now to explain what's actually going on.

Firstly, in US copyright law there's a special thing called "Fair Use". The idea of fair use is being able to allow people to fairly use copyrighted material in a way which does not affect the copyright holder's ability to capitalize on what they've made. Educational pieces, parodies, even a review, all fall under fair use in most cases.

HOWEVER, the tricky part is that many big-name companies have been pushing for stricter copyright laws which require services such as YouTube to immediately remove material which MIGHT infringe on copyright, fair use or otherwise. So what ends up happening is YouTube has an automated system in place which attempts to determine if you are indeed infringing on copyrighted material owned by a company which has put their material into the detection system. If a detection is made, what happens to the video depends on what the copyright holder wants to happen. Many will simply divert ad revenue away from your video (or start running ads to generate ad revenue for the copyright holder if you didn't monetize), a few will mute audio or sections of video which contain the infringing parts, but there's also a small number of copyright holders who will ban videos outright if a detection is made.

To help comply with fair use, the system is designed not to make a detection unless the detected material is in excess of a certain length without significant modification, I believe 30 seconds but I try not to ever exceed 25. Doing a voiceover overtop of a music track for instance, with the voice being substantially louder than the music will be recognized as modified enough to not trigger a detection regardless of length.

HOWEVER (again), there are a handful of large corporations who police their own material, (the traditional way of handling copyright), mostly television production companies and a small number of game developers. These corporations don't care if your stuff falls under fair use or not and will actively issue takedowns of things which even remotely look like they're infringing on their rights. If you REALLY feel like you're being bullied and that your content properly falls under fair use, you can fight these takedowns, but unless you're 100% positive and have spoken with a lawyer about it you really don't want to go through that, especially since it could end up going to court.

Now, in terms of LPs, something to consider is that the content of an LP is a mix of copyrighted material and original material, and thus falls under a massive grey area. The graphics you see and the music and sound playing in the background, that's copyrighted material, but any voiceover and gameplay actions, that's original material. Generally speaking, if you're going to do an LP, you want to follow some basic rules to avoid getting hit with copyright infringement stuff, fair use or otherwise:

1. Do a voiceover. If all you do is play the game and nothing else, you'll have a very hard / impossible time qualifying that as fair use. :P

2. Don't show cutscenes in their entirety. Only show bits and pieces so people know they're happening.

3. Learn to cut down your material and don't have long segments of no voiceover. Depending on the game, you may trigger automatic detection of copyrighted material thanks to the background music if it goes for long enough without you speaking.

4. Try to limit yourself to games which are owned by companies which ENCOURAGE people to talk about their stuff, which is actually the majority of them. A small handful of companies (notably Nintendo) take a middle-ground approach where they require you to follow certain guidelines and will still require you to hand over ad revenue to them, but will otherwise not prevent such videos from staying online. Only a REALLY tiny number of companies will actively try to prevent any and all videos being made of their games because, quite frankly, doing takedowns like that generates seriously negative press for them.

So yeah, follow those rules and you should be good.

I'm in Canada and in our copyright law we don't have fair use, but we do have "Fair Dealing", which is ALMOST the same thing, with some notable differences, one of which being that sources of copyrighted material need to be credited, which is why I have credits run in the videos I make. :B

Too bad. I actually prefer watching Let's Play videos without voiceovers. One, sometimes I want to hear the game sounds/music. Two, some people's voices aren't suited for voiceovers. Three, too many youtubers swear gratuitously which annoys me to no end. I end up turning the sound off, which of course, mutes the game sounds/music too, which leads me to search a different Let's Play.

The only voiceovers on games I watch generally are either reviewers, or those I watch live on a stream, often of the not-edgehard kind.