Li+ is fairly obvious, the 1s electrons do not shield each other well so each is attracted to an effective nuclear charge approaching 3 resulting in a very high IP. Be+ is different, the 1s2 2s1 configuration, isoelectronic with Li atom, has a well shielded 2s1 electron. The effective nuclear charge approaches 2 and the distance is greater. The result is a large second IP but less that of Li+ [and much greater than Li atom]. B+ is similar to Li+ in that the 2s2 electrons do not effectively shield each other, but they are shielded strongly by the inner 1s2 electrons resulting in an effective nuclear charge approaching 3 at a greater distance than Li+. This results in an IP less than Li+ but greater than Be+.
The reason that inert gas configurations or filled orbitals are of lower energy is not magic. Electrons added to the same orbital or to orbitals of the same [or very close] energy do not effectively shield each other from the nucleus. The increased nuclear charge means each electron is more strongly attracted to the nucleus, is of lower energy. These effects are very evident in the first and second periods and become more subtle at higher atomic numbers.