Why the phrase "That this House do now adjourn" contains "do" and not "does"?

The motion is not yet passed, so the adjournment is only a possibility. This means that the subjunctive form of do is used.

A verb is in the subjunctive mood when it expresses a condition which is doubtful or not factual. It is most often found in a clause beginning with the word if. It is also found in clauses following a verb that expresses a doubt, a wish, regret, request, demand, or proposal.

In English there is no difference between the subjunctive and normal, or indicative, form of the verb except for the present tense third person singular and for the verb to be.

The subjunctive for the present tense third person singular drops the -s or -es so that it looks and sounds like the present tense for everything else.

English Plus

All these conditions apply to the sentence’s use of do, which at first appears to be a substitute for does but is really the subjunctive form of the verb do.