Links are nothing but character strings. They are the URL's given above, plus a few bytes of surrounding HTML to note them as links and the text that actually gets clicked on.
They do not consume any resources beyond that used to send any other string of characters a few dozens of characters long. And they use the exact same resources whether they point to an address on your site or any other site.
That having been said, the PHP and MySQL processing which has to take place to get that info and then echo it it out WILL take longer and use more resources the more 'complex' the page is.
But in no way does every link require a database call. That's completely absurd and the sign of someone not at all familiar with how things work.
For instance, click 'Forum' to go to the Tinyportal Boardindex. Down at the bttom it tells you exactly how many queries were required. There are 30 links just between the Recent Posts listing and the 'Upcoming Birthdays' list in that bottom-of-page Info Center. Yet everything on the whole page only requires around 25 queries (this can fluctuate a bit, but that's what it showed when I checked).