This is the grave of Shams-e Tabrizi, about whom the Enc. Britt. Has: “The decisive moment in Rûmî's [that is: Mevlana – D.O.] life occurred on November 30, 1244, when in the streets of Konya he met the wandering dervish—holy man—Shams ad-Dîn (Sun of Religion) of Tabriz, whom he may have first encountered in Syria. Shams ad-Dîn cannot be connected with any of the traditional mystical fraternities; his overwhelming personality, however, revealed to Jalâl ad-Dîn the mysteries of divine majesty and beauty. For months the two mystics lived closely together, and Rûmî neglected his disciples and family so that his scandalized entourage forced Shams to leave the town in February 1246. Jalâl ad-Dîn was heartbroken; his eldest son, Sulṭân Walad, eventually brought Shams back from Syria. The family, however, could not tolerate the close relation of Jalâl ad-Dîn with his beloved, and one night in 1247 Shams disappeared forever. It has recently been established that he was indeed murdered, not without the knowledge of Rûmî's sons, who hurriedly buried him close to a well that is still extant in Konya.”
I suppose (in view of this disappearance) that the “grave” is a monument rather than an actual grave. Near the mosque stands a heavily restored türbe with a peculiar ornamentation on the roof.