I remain puzzled on how this affects things. It is a rule by Executive, correct? How is the Senate involved if Trump can do a new one to over-ride?
If the Senate had passed the resolution (within 60 days), and Trump had signed it, the rule would have been dead, and the BLM could not bring it up again or craft a significantly similar rule to replace it.
Now the rule has to be replaced by the BLM, or it will stand. Replacing the rule means redrafting the rule, having comment periods, hearings, etc. which will take time (several months, minimum). In the meantime, look for drilling on federal land to drop off fast, and drilled wells which have no feeder pipelines to gas processing plants will have to cease or limit production. As a result, the Government will not get royalties, wells may be damaged, etc.
The BLM and other Federal Agencies also oversee the permitting process for the feeder pipelines, and their delays have made the rule about methane an issue. They created or exacerbated the 'problem'. The oil companies would often like to be collecting revenue on that byproduct rather than flaring it onsite.
If you want a peek into the liberal abyss, comments on the Hill website about the rule were foaming at the mouth insane, babbling about 'toxic gas' (methane) polluting "everything" despite the remoteness of most locations on BLM land, and full of gratuitous denigration of "evil oil". Chances are the concentration of methane is higher in their hybrid car during their commute from biological sources inside the vehicle cabin than from any oil well on federal land, especially if they car pool.