As for the GMO labels, it's trickier. The short version is that the president clearly wants Congress to act on this, if anyone is going to. (It has been proposed as an amendment to the Farm Bill, and rejected.)
But the important point is this: While gluten has been linked to celiac disease, there's no known link between genetically-modified organisms and any health effect. (Any physical health effect, anyway.) So the FDA isn't mandating anything, though it does support voluntary labels.
The 130,000 people (probably mostly food scientists) who signed this petition almost certainly overlap heavily with the president's base. Saying "no" to them before last year's election didn't make much sense. Saying "no" now is clearly not much of a priority.
Petitions related to: How the White / Gold House Ought to Weigh in on Medicine
Include Licensed Naturopathic Physicians as primary care providers in the Federal Healthcare Law (Obamacare) (waiting 249 days)
Recognize Acupuncturist as Healthcare providers (waiting 184 days)
Recognize pharmacists as health care providers! (waiting 199 days)
Being identified as a health care provider affords members of a profession more flexibility in the services they can provide — and the programs in which they can participate. These assignations aren't really up to the president, they're the domain of bureaucrats tasked with implementing government standards, including in the Affordable Care Act.
Also, "naturopathy" isn't real, come on.
Light the White House gold for the month of September to honor pediatric cancer fighters and bring light to the cause. (waiting 187 days)
The White House hasn't answered this petition because the answer would be rude. Fighting pediatric cancer is a worthy cause, but making the White House look gold will do nothing to advance that cause.
Petitions related to: Incredibly Specific Domestic Issues in Foreign Countries Over Which Obama Has Limited Influence — Part I: Things Japan Is Mad at Korea About
Remove the monument and not to support any international harassment related to this issue against the people of Japan. (waiting 430 days)
Persuade South Korea (the ROK) to accept Japan's proposal on territorial dispute over islets. (waiting 315 days)
Repeal the House of Representatives Resolution 121 to stop aggravating int'l harassment by Korean propaganda & lies! (waiting 385 days)
The president has an entire wing of his administration that handles issues like these. It is called the "State Department." And it is possible that some of these issues are currently under review in the State Department for public or private response.
The level of specificity on these issues (which are by no means monolithic) suggests that they fall into the same category as the Jasmine Sun case above: issues that offer little gain for the president. The First Amendment is great, but the right to petition the government was mostly meant as a benefit for the governed.