The flanker, who has scored two tries in 60 appearances for the Welsh club, has signed a two-year deal at Irish.
He said: "I have thoroughly enjoyed my time at Ospreys but feel it is time to move on and I am keen to do this at London Irish.
"There are a few Welsh lads in the current London Irish squad who have gone on to be extremely successful in recent seasons and I hope to follow suit when I join this summer."
London Irish head coach Toby Booth said: "Tom's ability to play at both blindside flanker and number eight means he will add further to our back-row options. He is a top class defender that possesses very good set-piece attributes."
Smith follows George Skivington in signing for Exiles but the news comes just a day after Paul Hodgson confirmed he is leaving to join Worcester Warriors.
*** SEE THURSDAY'S CHRONICLE FOR THE LATEST LONDON IRISH NEWS, INCLUDING AN INTERVIEW WITH JONATHAN JOSEPH ***