There are a number of minor fast days that Liberal Jews tend not to observe (e.g. 17th Tammuz, 10th Tevet – fast days associated with the history of the Temple in Jerusalem. Tish’ah b’Av – another Temple-related fast day also tends not to have a place in Liberal Jewish observance. In early Liberal Judaism, Purim also tended to be ignored, because it was based on a triumphalist, violent story that almost certainly didn’t happen – though recently it has entered the Liberal Jewish calendar.
The essence of Liberal Judaism is ‘informed choice’. So, in theory, every Liberal Jew observes each festival and Shabbat according to his or her own individual choices and preferences. So observance will differ from one Liberal Jew to the next as she or he chooses from the range of possibilities available at and unique to each festival (e.g. lighting candles the evening the festival commences, eating the specific food relating to that festival, attending synagogue for the services etc).
The festivals that have their origins in the Torah are specifically related to the phases of the moon. In particular this applies to Pesach, which falls on the first full moon of the spring, and Sukkot, which is exactly six lunar months later. The other Torah-based festivals are all connected to those two (Shavu’ot is exactly 50 days after the second day of Pesach, Rosh ha-Shanah is 15 days before Sukkot and Yom Kippur is 10 days after Rosh ha-Shanah. In order for the first day of a new lunar month to be declared, two independent witnesses had to report its sighting to the rabbinic authorities in Jerusalem. Depending how late in the day these reports were received, the first day of the new month might be 29 or 30 days after the previous one. Once Jews moved away from Jerusalem, it became impossible for that information to be relayed to distant communities in time. So a second day of each major festival was introduced – the first day was based on the assumption of when the new moon would be declared and the second just in case its sighting in Jerusalem was delayed, making everything a day later. So in the Orthodox Jewish calendar outside of Israel there are 2 days for the start of Pesach, for Shavu’ot, for the start of Sukkot and for Rosh ha-Shanah (though, thankfully, not for Yom Kippur!)

Nowadays, of course, the timing of the new moon can be calculated precisely, so Liberal Jews do not feel the need to maintain this ancient tradition. Orthodox Judaism, however, is ever reluctant to remove practices once they have been established, so two days are still observed in Orthodox communities outside of the land of Israel.