The Toffees failed to find the win which would have kept the pressure on the Premier League's top six.
They climbed to eighth, above Arsenal, to move two points behind fifth-placed West Ham, but Everton's inconsistencies look like costing them a European spot.
The game at least represented a return for Villa skipper Jack Grealish, who ended his three-month injury lay-off as a second-half substitute.
The skipper was unable to conjure a winner, though, as the hosts remain 11th with three games to go.
The game never inspired and only Bertrand Traore's tame effort and Calvert-Lewin's header punctured a sluggish start.
The game was played at pre-season pace as both sides laboured and struggled to find anything resembling momentum.
Mings, though, should have at least hit the target when he wasted Villa's best opening after 17 minutes.
Ross Barkley's corner was only cleared as far as Douglas Luiz, who crossed for an unmarked Mings to head wastefully wide from seven yards.
Everton, with two wins in their last previous league games, needed victory to move into the top six yet they appeared aimless, with Calvert-Lewin and Richarlison relegated to bit-part roles.
Without the banned Ollie Watkins Villa were just as toothless and would have already been looking to Grealish on the bench for imagination.
The second half at least offered more promise when Everton created an early chance.
Gylfi Sigurdsson's deep free-kick found the stretching Godfrey and Martinez had to be alert at his near post to turn the ball behind.
The Toffees saw more of the ball but again failed to really stretch Villa in open play, with the hosts also relying on set-pieces to threaten.
Jordan Pickford beat away Anwar El Ghazi's free-kick after an hour just as Grealish prepared to clamber off the bench.
Andre Gomes fired over after working space on the edge of the box and the game was given a desperately needed lift when Grealish replaced Traore with 18 minutes left.
But the England international failed to find some magic and it was left to Martinez to ensure Villa claimed a point with three minutes left.
Lucas Digne's deep cross picked out Calvert-Lewin and the goalkeeper was equal to his header, with Abdoulaye Doucoure and Richarlison unable to convert the rebound.