According to the Cambridge Dictionary, if someone has a sweet tooth, (s)he likes eating sweet foods and meals. So, what's the word for who is interested in salty ones?
There are some usages of the expression “salt/y tooth” which does not appear to have become popular yet. Probably you can generally refer to this category of people as “salty food lovers”:
Oregano Can Satisfy Your "Salt Tooth" (Time.com)
Yes, There Is Such A Thing As A ‘Salt Tooth’ (Huffinton Post)
Salty tooth (UD)
“Salt tooth” is a real thing, and we feel so validated” (hellogiggles.com)
Do You Have A Salt Tooth? (naturalon.com)
There is some evidence for the use of 'savoury tooth', which is often used in contrast to 'sweet tooth' as it is a clearly parallel formation.
'Savoury tooth' and 'savory tooth' combined turn up about 280k Google results, but I can't get either to show up in an ngram.
As a general observation the Google results seem to show a bias towards magazine articles, blogs and other social media. 'Savoury tooth' also turns up a company making savoury protein bars.
My take from this would be that 'savo(u)ry tooth' is a phrase that people will understand, and that they use, but is often used in direct contrast to 'sweet tooth' and probably gets invented anew when people are casting about for a term.
Afterthought: It occurs to me that I have, possibly in a noticeably British way, defaulted to the use of 'savoury' as an antonym of 'sweet' and answered in that vein. However, @Eilia was very specific in using 'salty', so I may have veered off track with this answer.
I've always been a bit foggy on the distinction between 'salty' and 'savoury' since I cannot imagine a savoury flavour that does not have a salt element and possibly I think of savouriness as an assemblage of flavours rather than a distinct category.
In poking around the subject I found this wordforums discussion which suggested that British people tend to refer to things as 'savoury' where US English speakers would use 'salty'. Which would explain why US You Tubers' first reaction to Marmite is always to say 'it smells so salty', when I think Marmite smells of savoury/umami goodness, whereas salt doesn't smell (is that just me?)
Stitching together a few Greek roots is often enough to coin terms people in medical and biological professions are likely to be able to interpret: "halophile". It turns out that Halophile is an actual biological term for organisms thriving in salty environments. For a tongue-in-cheek pseudo-learned expression this should still (or even particularly well so) serve reasonably well.
The terms addict and junkie are listed in dictionaries, extremely common and easily understood, and used in a myriad of combinations.
If the OP really wants a single word, I googled and found salt-aholic or saltaholic, the latter is a newly submitted word in the Collins Dictionary which is awaiting approval, which kinda tells us it's a term not commonly heard or used.
Google Ngram shows results only for the last two expressions in the following list: savory tooth, savoury tooth, salt tooth, salt junkie, salt addict (blue line), sugar addict (red line).
Taken from the CNN Health article Love salt? You might be a 'supertaster'
If you love salty snacks and reach for the saltshaker like clockwork at every meal, you might think you have dull or underpowered taste buds that need a boost to get excited.
In fact, just the opposite may be true: A new study suggests that you may love salt because you're a "supertaster" -- a person who experiences tastes such as saltiness and bitterness more intensely than other people do.
The article goes on to say:
Hayes and his colleagues were surprised to discover that the supertasters liked more salt rather than less, even though they were more sensitive to it.
For instance, salt helps cancel out bitterness, one of the sensations that supertasters experience in Technicolor. This may explain why the supertasters in the study perceived the low-sodium cheddar cheese to be twice as bitter as the Cracker Barrel, and liked it far less than the other study participants did.