Probably is a good idea if he wears pants of some sort.

My DH usually wears a pair of khaki shorts with a polo or button-up short sleeve collar shirt with his gym shoes and we've been to every signature but V&A.
After eating at a number of the signature restaurants, I was quite surprised at how unenforced the dress code was. Personally, I always think it's nice to make an effort when you visit these places. On a previous visit to The California Grill, I wore a casual shirt (untucked and no tie,) with some smart dark coloured jeans and shoes. I felt overdressed. Sure, there were people wearing formal attire there, but they were few and far between, with most of the men at least wearing t-shirt and jeans with trainers.
I think its a personal choice, but as long as you aren't wearing shorts or t-shirts with offensive graphics on them you are pretty safe.
To paraphrase:
. . . Dress Code? We don't need no sticking dress code!
1) Is there a dress code for signature eateries?
2) Yes.
3) Is it enforced?
4) With exception of V&A, no.
5) Unless really obscene, almost anything goes.
Both also had on the cheap rubber flip flops. Despite their arguements that they were dressed appropriately they were turned away. The second time, I saw a couple turned away at Yachtsman. The woman had on a very pretty sparkly top, high heel sandals, and a pair of dark blue jeans that were ripped and were full of holes. When told the jeans were not appropriate attire, she got really irate and yelled out, "Do you know how much these jeans cost? probably alot more than you make in a night!". A manager came running out and another arguement insued. The manager did not budge though and I heard him say he was sure the dress code was explained when they booked the reservation and any clothing were holes in them were not allowed, no matter if they were expensive and the holes had been placed in them intentionally.