Brown has established as Doncaster's first-choice tighthead prop after making the switch from Old Albanians, while Yeandle has made 27 appearances at hooker this season after joining from UWIC.
However, former Saracens prop Brown and Exeter-born Yeandle, who are both 22, will now head to Sandy Park after the RFU Championship play-offs.
Doncaster director of rugby Brett Davey told the club's official site: "It's disappointing to lose two quality players, but I would never begrudge a player moving to the higher level.
"I would like to take this opportunity to wish them the best of success with Exeter in the Premiership and the future rugby careers."
Davey has moved to replace Brown and Yeandle by signing Yorkshire Under-20s front-rowers Tomas Francis and Adam Tamanis.
Leeds University student Francis initially arrived at Castle Park on a development contract in January but the 19-year-old tighthead has now earned a new deal.
Loosehead prop Tamanis, who is also 19, will make the step up after playing for amateur side Doncaster Phoenix.
Davey added: "Both players have tremendous potential and the fact that they are both local lads is very encouraging.
"I am looking to lads in the Yorkshire region for the future of this club and these two are prime examples and represent the start of our ambitions."
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