In Chamber of Secrets, Tom Riddle mocks the assistance Harry receives at the start of their final confrontation - Fawkes and the Sorting Hat arriving - as pathetic.

So this is what Dumbledore sends his great defender. A songbird and an
old hat.

It's established elsewhere in the series, however, that Riddle was a top student, and that he was incredibly well versed in lore related to the Hogwarts founders. Wouldn't he have known exactly how valuable a phoenix could be, in that situation? And how potentially dangerous the hat was to his plans?

Tom / Voldemort is incredibly arrogant, of course. But this seems hard to explain even with arrogance. Wouldn't he have killed Fawkes and destroyed the hat before dealing with Harry, who didn't even have a wand at that point?

Tom forgot phoenix tears can heal.

Tom Riddle didn’t remember that phoenix tears could heal at the time he was confronting Harry. He says himself when he realized what had happened that he forgot that.

“Phoenix tears …’ said Riddle quietly, staring at Harry’s arm. ‘Of course … healing powers … I forgot …”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)}}

As for why he didn’t think the Sorting Hat was more of a threat, because it’s known that true Gryffindors can pull the Sword of Gryffindor out of the Hat, that’s not mentioned. It’s possible he didn’t think of that or forgot that as well at the time.

Neither Fawkes nor the hat appeared to be a threat.

Phoenixes are not known for their offensive prowess. There wasn't really anything that Fawkes could do to harm Tom Riddle. At most, Fawkes would be able to heal Harry if he got injured, but Riddle was planning on killing Harry rather than injuring him.

As for the hat, prior to this event there does not seem to be any record of the hat doing anything other than sorting students. As such, Riddle had no reason to think that it was anything other than a useless hat.

Since Riddle had nothing to fear from Fawkes or the hat, he had no reason to deal with them. He had a wand and a basilisk and he was up against an unarmed second-year student. He could well afford to play around. In fact, Riddle probably wanted Harry to have these items to help him, as that would be a better display of Riddle's superiority. Note what he says right before calling the basilisk:

"Now, Harry, I'm going to teach you a little lesson. Let's match the powers of Lord Voldemort, Heir of Salazar Slytherin, against famous Harry Potter, and the best weapons Dumbledore can give him...."

This is quite similar to what he did with Harry at the end of Goblet of Fire, when he gave Harry his wand. He didn't want to simply kill Harry; he wanted to demonstrate his superiority over Harry. As he said then:

I will give him his chance. He will be allowed to fight, and you will be left in no doubt which of us is the stronger.

That Fawkes ended up blinding the basilisk was not the biggest deal, as the basilisk could still kill with its fangs. Once the basilisk struck Harry, Riddle probably should have kept Fawkes away from him, but as Bellatrix pointed out he wasn't thinking of the fact that phoenixes have healing powers. And even after Harry was healed there still was no real issue because Riddle still had a wand with which he could kill Harry, while Harry only had a sword.

The real mistake was leaving the diary lying around where it could easily be destroyed, but that too only became a problem because Harry unexpectedly pulled a sword out of the hat which ended up getting him access to a basilisk fang with which to destroy the diary.

Perhaps Voldemort should have respected Dumbledore enough to think that there might be some purpose to an apparently useless hat, but that's where the arrogance factor comes in.