c. 1600, "to free from obligation;" 1630s, "to refuse or neglect to oblige," from French désobliger (c. 1300), from des- (see dis-) + obliger, from Latin obligare "to bind, bind up, bandage," figuratively "put under obligation," from ob "to" (see ob-) + ligare "to bind," from PIE root *leig- "to tie, bind."
Colloquial sense of "put to inconvenience" is from 1650s (implied in disobligingness). Related: Disobliged; disobliging; disobligingly.