The torah commands the Israelites not to "work" on Shabbat or on festivals (here, Pesach). The torah does not define what this means; that was expounded in the oral law, which was transmitted (orally) from teachers to students and finally written down after the destruction of the temple, probably around 200CE. The mishna attributes the arguments it records to rabbis who lived between about 100BCE and 200CE, so while we can't say for certain exactly what the practice was circa 30CE, the odds are that the mishna is not far off.
Sealing the cave
The mishna articulates 39 categories of labor that are forbidden on Shabbat in the Babylonian Talmud, Tractate Shabbat 73a (this is the Soncino translation):
The primary labours are forty less one, vis: sowing, ploughing, reaping, binding sheaves, threshing, winnowing, selecting, grinding, sifting, kneading, baking, shearing wool, bleaching, hackling, dyeing, spinning, stretching the threads, the making of two meshes, weaving two threads, dividing two threads, tying (knotting) and untying, sewing two stitches, tearing in order to sew two stitches, capturing a deer, slaughtering, or flaying, or salting it, curing its hide, scraping it (of its hair), cutting it up, writing two letters, erasing in order to write two letters (over the erasure), building, pulling down, extinguishing, kindling, striking with a hammer, and carrying out from one domain to another: these are the forty primary labors less one.
For the purposes of this question the interesting items on this list are: ploughing, building, possibly kneading, possibly striking with a hammer, and carrying. (Depending on what any assigned guards were to do if confronted, more items on this list might apply.)
The g'mara, written down c. 200-500CE, expands on ploughing thus. A tanna is a rabbi who is contemporary with those of the mishna:
A Tanna taught: Ploughing, digging, and trenching are a]l one [form of] work. R. Shesheth said: If one has a mound [of earth] and removes it, in the house, he is liable on the score of building; if in the field, he is liable on the score of ploughing. Raba said: If one has a depression and fills it up: if in the house, he is liable on account of building; if in the field, he is liable on account of ploughing (73b).
Filling the spaces between the stone and the cave mouth seems like filling a depression.
Depending on how the stone was to be sealed (I'm imagining something like mortar but I don't know), there are other violations. The g'mara holds one liable for boiling pitch or for making an earthenware vessel (74b). The prohibition on "building" covers tasks like laying bricks and sealing casks, which might apply to sealing the space between the stone and the cave walls. If the mortar had to be mixed on that day, that would violate some of the "cooking"-related items like kneading, sifting, etc. (While these labors are based on preparing food, they are not only about food.)
The materials to perform this work would also need to be carried to the cave. Unless everything was already in the public domain (e.g. in the street, rather than in a building or courtyard), carrying a usable quantity (enough to seal a bag, per mishna on 80b) would be forbidden absent an eiruv. Tractate Eiruvin explains in great detail how to construct a boundary around a space (such as a group of houses, or even a town); with that affordance you can carry anywhere within that space. We don't know if an eiruv was in place in this time and place, but it seems unlikely to me that it would have encompassed the burial cave. (Generally it encompasses residential areas, and there would need to be a physical boundary.)
Further, a mishna on 75b continues:
They also stated another general principle: whatever is fit to put away and such is (generally) put away, and one carries it out on the Sabbath, he is liable to a sin-offering on its account. [...]
In other words, put your weekday tools away and observe Shabbat.
Setting a guard and asking Pilate for a ruling
A person is free to move about on Shabbat (within a town and its environs), so having somebody stand watch at the grave is not inherently a problem. (It's questionable because by doing so you are probably not celebrating Shabbat, but it does not directly violate one of the 39 categories of primary prohibition.) Of course, because of carrying (above), that guard would not be able to have a weapon.
Modern Jewish practice forbids discussing business and other "mundane" affairs on Shabbat. I cannot find anything that speaks to this in the mishna (though I could have missed it, of course -- the mishna's pretty big).
So it appears to me that (a) going to Pilate, assuming he was locally resident, and (b) having someone watch the grave would not necessarily be violations of the law. Given the urgency the Pharisees apparently felt, it is not unreasonable that they would have availed themselves of any leniencies in the law.
Shabbat versus festivals
Because the torah uses the same language (melacha) to refer to labor that is not to be done on both Shabbat and festivals, the rabbis understand the restrictions to be basically the same. The only leniency is that on a festival (that is not on Shabbat), one is permitted to do certain actions for the sake of the holiday, such as adjusting a previously-lit flame to cook your food or carrying items needed for the festival between domains. None of that would seem to apply to the case at hand.
The Pharisees could probably, without violating the law, have the discussion with Pilate and send somebody to watch the grave. They would not have been able to seal the grave, on either Shabbat or the festival, without violating several categories of forbidden labor.
Please note: This answer was written for a neutral, academic audience
and is not intended to be interpreted in the context of a religious
belief or doctrine.