The shot-shy Cherries missed the chance to put clear daylight between them and the chasing pack and have now taken just two points from their last three outings.
Boro boss Chris Wilder made a big call by dropping goalkeeper Joe Lumley and handing Luke Daniels his first start since December.
But neither Daniels nor opposite number Mark Travers were seriously tested for much of a first half lacking in attacking quality.
Seven minutes later, Dominic Solanke saw his effort from 12 yards blocked by Isaiah Jones after good play by right-back Ethan Laird.
Boro then had two penalty appeals waved away by referee Tim Robinson in the space of a minute.
First, Adam Smith appeared to block a Jones cross with his outstretched arm and then Phillips looked to have played the ball against his own hand after mis-controlling it.
It took until the final minute of the half for the first attempt on target and it came from the visitors.
Marcus Tavernier found himself in space 30 yards from goal before rolling a pass into the feet of Duncan Watmore and his well-struck shot was pushed away by Travers.
The home side came close to breaking the deadlock nine minutes before half-time when Philip Billing fired a 25-yard free-kick just over the crossbar.
Bournemouth were buoyed by promotion rivals Nottingham Forest losing 1-0 at Luton earlier in the day.
But it was Boro who started the second half looking more likely to open the scoring, with Tavernier flashing a header wide from Marc Bola’s hanging cross.
Bournemouth thought they had taken the lead when Jefferson Lerma met Cantwell’s 58th-minute corner with a thumping header only to be denied by the underside of the crossbar.
The Cherries had another opportunity from the rebound, but Danish international Billing could not keep his volley down.
Bournemouth boss Scott Parker brought on substitutes Jaidon Anthony and Jamal Lowe in a desperate bid to break down Boro’s stubborn defence.
The pair combined in the 77th minute when Anthony floated in a teasing left-wing cross but the rising Lowe could only head high over the crossbar.
Anthony also curled a shot into the arms of the under-worked Daniels with Bournemouth’s first shot on target six minutes from time.
The final whistle was met with jeers from the home fans as their Boro counterparts celebrated a hard-fought point.