William Hartnell did small walk-on parts on stage in his first job as a teenager with Frank Benson's Shakespearean Company. No Shakespeare on IMDB though.
Jacqueline Hill was Lady Capulet in the 1978 BBC Romeo and Juliet.
Early in his career, Patrick Troughton did several Shakespeares, two of which were Hamlet - he was Horatio in a 1947 production, and rather more famously the Player King in Olivier's 1948 version. In the latter year he was also Edmund in a TV King Lear. A few years later he was back with Olivier to be the murderous Tyrell in the 1955 Richard III.
The closest I can get to a Shakespeare connection for Jon Pertwee is his appearance as the Soothsayer in Carry on Cleo.
Caroline John was Hero in a 1967 BBC production of Much Ado About Nothing.
Tom Baker's film debut was in a 1968 release of The Winter's Tale, as the Bear, which seems a bit inglorious.
Lalla Ward was Ophelia to Derek Jacobi's Hamlet in the 1980 BBC production.
Peter Davison has published commentaries on a number of Shakespeare plays, including the First Quarto version of Richard III, Henry IV Parts I and II, Othello, Richard II and Henry V, though he is better known as a scholar of George Orwell. I refer of course to the former Professor of English at De Montfort University in Leicester, not to the actor whose real name is Peter Moffatt and who doesn't appear to have any Shakespearean credits.
Colin Baker has done a fair amount of stage Shakespeare but nothing on screen.
Sylvester McCoy plays the Fool (so not much new there) in the recent Trevor Nunn / Ian McKellen King Lear.
In the same year as he became the Eighth Doctor, Paul McGann was Bassanio in Channel 4's The Merchant of Venice.
Christopher Eccleston was Ben Jago in Andrew Davies' 2002 update of Othello.
Billie Piper was Hero in David Nicholls' 2005 update of Much Ado About Nothing.
David Tennant has done a lot of Shakespeare on stage, and I've been enjoying his performances on the Arkangel series of audios as well - Henry VI, Antipholus of Syracuse, Laucelot Gobbo, Edgar, Mercutio, and I'm looking forward to him as the Porter in Macbeth which is next on my list. Apparently his Hamlet is coming out on DVD; I can't wait.
Not aware of any Shakespeare action from Matt Smith; no doubt he's done a fair bit but it hasn't hit his professional CV.
This is all leading somewhere, just not quite sure where at the moment...