This is a follow-up question to Why did Aunt Petunia mistreat Harry Potter given that Harry was so rich and she had more to gain by treating a rich boy well?
One reason Aunt Petunia mistreated Harry was that she did not know he was rich. If she knew, she would naturally have treated this golden goose nicely. Her retirement would be secured if Harry financially supported her in old age.
Why didn't Harry tell Aunt Petunia he was rich? Surely every boy wants to be treated nicely by his family.
Before Hagrid's visit, Harry wasn't even aware he had money at all.
"Um -- Hagrid?"
"Mm?" said Hagrid, who was pulling on his huge boots.
"I haven't got any money -- and you heard Uncle Vernon last night ... he won't pay for me to go and learn magic."
"Don't worry about that," said Hagrid, standing up and scratching his head. "D'yeh think yer parents didn't leave yeh anything?"
"But if their house was destroyed --"
"They didn' keep their gold in the house, boy! ...
(PS, Ch. 3, "The letters from no one")
After that, he still didn't realize just how rich he was till the visit to Gringotts:
Griphook unlocked the door. A lot of green smoke came billowing out, and as it cleared, Harry gasped. Inside were mounds of gold coins. Columns of silver. Heaps of little bronze Knuts.
"All yours," smiled Hagrid.
All Harry's -- it was incredible.
(PS, Ch. 5, "Diagon Alley")
Even after he found out he was rich, he couldn't tell Dursleys because he knew they would just steal his money, NOT be nice to him hoping he'd give the money to them:
All Harry's -- it was incredible. The Dursleys couldn't have known about this or they'd have had it from him faster than blinking. How often had they complained how much Harry cost them to keep? And all the time there had been a small fortune belonging to him, buried deep under London. (PS, Ch 5 again)
...
He had never mentioned his Gringotts bank account to the Dursleys; he didn’t think their horror of anything connected with magic would stretch to a large pile of gold.
(CoS, Ch4, "At Flourish and Blotts")
... And, even if they did, they couldn't use it in a Muggle society.
Harry said nothing. He felt a bit awkward. Stored in an underground vault at Gringotts in London was a small fortune that his parents had left him. Of course, it was only in the wizarding world that he had money; you couldn’t use Galleons, Sickles, and Knuts in Muggle shops.
(* CoS, Ch4, "At Flourish and Blotts"*)
Harry didn't know he was rich either.
Harry lives for 11 years under the impression that he is poor. Not until the event of the first book, when Hagrid shows him the vault of gold his parents left him, does he understand that he has money. In book 2 we see Harry employing his family's fear of magic to receive better living conditions during the summer.
I don't think Harry would have told them after he found out about the gold in the bank vault, because the Dursleys would have quite likely tried to pressure him into giving them some of the money. Vernon especially seems to begrudge any money spent on Harry and might believe he owes them for the years spent taking care of him.
