Providing fake info during domain registration - does it matter?

ICANN (not the domain registrar) requires that all information in your registration be valid.

If any dispute arises (see the ICANN Uniform Domain Name Dispute Resolution Policy rules here) you will be contacted via the means specified in your domain registration. Notice that section 14 of the rules is a section that defines what happens as part of a 'default' (in other words, they can't contact you): They'll proceed with a judgement, and you won't get a say in the proceedings.

ICANN has the power to take a domain from you and give it to somebody else.

So yes, it's important that you include valid information in your registration information.

For a comparison of a what a private domain registration looks like (compared to a regular domain registration) see this comparison: public vs. private.

Personally, I wouldn't insert false data. Whilst you generally should not have problems, it is usually against the rules of the registrar you are using and for some TLD's, it is against the registry's rules.

If someone chooses to complain (for any reason) and they have no way to find out who owns the record, and they kick up enough fuss, it is possible that either the registrar or the registry will delete the domain.

However, if privacy is important for you, some places offer privacy for free.

Yes, it matters!

At most registries it is a breach of their contract to supply incorrect information, and if caught they can cancel your domain name.

If the registration is for personal (non-trading) use then some registries (like .uk) offer an opt-out so that personal information is not revealed in the "whois" database. However the registry itself still requires your real information.