This whole lawsuit thing is just to gin up the base to increase turn-out in November and to give them something to use to obtain money from their corporate backers and contributors. I think they are waiting for the results of the Nov. elections -- if they gain control of the Senate, then impeachment will be the first order of business and they won't stop. If they fail to take the Senate, then they will pursue this lawsuit with vigor.
There is no legal precedence for a suit against a sitting President by a branch of Congress, not to mention the fact there there are no laws that have been broken and the House, having suffered no financial loss, lacks standing to bring the case in the first place.
No, this was done entirely to garner votes from the wacked-out limb of Repubs. It tweaks their inherent hatred of the Black President who also isn't a Republican and seeks to punish him for not "keeping in his place" and for defeating TWO of their own candidates. You'd think that after all their lies, manipulation of public opinion, diluting minority and Democratic voters and out-right cheating at the polls, they still lost the last two national elections, they would do some self-reflecting to figure out that their "message" just isn't resonating with the majority of Americans.
Their immediate goals are two-fold: teach the next uppity Black man/woman to stay in his/her place, and prevent another Democrat from being President. They will not stop until they have achieved their ultimate goal of a christian theocracy, one-party rule and the destruction of the middle-class. If they are ever again in control of the House, the Senate and the Presidency, say good-bye to social security, medicare, ACA health insurance, food stamps, highway bridges any anything else that might bring aid and/or comfort to the masses. All you have to do is listen to them. They are telling us what they want -- we just have to listen.