Booking hotel with my credit card for a third person. Will that person have issues at check-in?

There are two different steps:

1. the payment of the vacation where you can mostly use any kind of payment eg. credit card, bank transfer, voucher etc

2. during the stay it is probably that the hotel ask to the customer its credit card, not for the payment but for an insurance

Only in the second step is necessary that the customer has its own credit card.

Almost certainly. The hotel doesn't care who's paying as long as somebody is paying, and that somebody has enough credit to cover the hold they'll put on the card at checkin.

If you're really worried, just give them a call and check.

Actually using your card for a different person who would check in at the property is a bit risky, but you can actually just do that as long as you are going to get in touch with the property the soonest possible time to advise them that the person who check in will use a different card instead to pay the reservation so they can change the credit card number on file.
Safest way to make the call to the hotel after you booked a reservation is at least more than 72 hours for the refundable type of reservation.

But if you're trying to choose a non refundable reservation to get the lowest cost I suggest you think it over not just twice because even 3rd party reservations were not given info when the hotel will try to run your card to see if it works.

And another thing, there's also a refundable type of reservation that requires a deposit on a certain length of time, so make sure to read the policy 1st on each room you choose.

I tried to do this once and discovered that hotels are very strict on this issue. Hotels don't care who pays for the room itself of course, but every major hotel chain in the world requires that the person staying at the hotel must have their own credit card in their name which the hotel would physically swipe on site for an incidental deposit, regardless of who made or paid for the reservation.

A third party cannot contact the hotel and give them their credit card number over the phone for the incidental deposit, for example. The person whose credit card is used for incidentals not only has to be present, but also their name must be on the reservation.

Some hotels will allow a cash deposit, and perhaps there may be a small handful of hotels that do allow a 3rd party incidental deposit, but I've not found one that will allow it – certainly not the more established hotels and hotel chains. You'd just have to call ahead, speak with the front desk manager, and see if something can be worked out. But my experience has been that this policy is cut and dried and exceptions are rarely, if ever, made.