The Olympic poster boy and his partner Pete Waterfield were in with a chance of grabbing the host nation's first gold of the Games in the synchronised 10-metre platform event on Monday, but fluffed their fourth dive and wound up just outside the medals.
Daley, whose father Robert died from cancer last year, repeated a tweet from user @Rileyy_69 to his followers that said: "you let your dad down i hope you know that".
The diver responded: "After giving it my all ... you get idiot's sending me this."
The user later apologised.
Police from the English county of Dorset said a 17-year-old man was arrested in the early hours of Tuesday at a guest house in the Weymouth area on suspicion of malicious communication.
"(He) has been issued with a harassment warning in connection to tweets to Olympic diver Tom Daley," said a police spokeswoman.
The teenager was released on bail while police investigate other communications on his Twitter account and he will return to the police station at a later date.
A harassment warning is not a criminal conviction but a notice that a complaint has been received and if it were repeated, those acts could amount to harassment.
Britain's chef de mission Andy Hunt said the tweets were "completely unreasonable and a threat" but said Daley was "unaffected".
Daley will compete in the individual 10-metre platform event starting on August 10.
"Everyone knows if you use social media extensively you get bad as well as good," Hunt told reporters.
"If you open the door people come through it as well as go out the other side and we have been very clear about that. We have put out a lot of guidelines and a lot of education about this."
Twitter, a microblogging site which allows users to send short messages to each other and advertise views to a potentially huge audience, has hit the headlines at the Olympics after some ill-considered tweets by athletes.
Swiss football player Michel Morganella was expelled from the Olympics on Monday for a tweet which the Swiss team said "gravely insulted and violated" the dignity of South Korea after his team's 2-1 defeat.
Greek triple jumper Paraskevi Papachristou was also withdrawn after a tweet which was criticised as racist.
But Twitter has also given a platform for athletes to protest against Olympic rules that they disagree with. Some U.S. athletes have posted tweets against restrictions on the promotion of non-official sponsors.
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