Can any plant regenerate missing tissue?
In general, plant cells only undergo differentiation at special regions in the plant known as meristems. Two of the primary types of meristem are the root apical meristem (at the tips of roots) and the shoot apical meristem (at shoot tips)^. Within the shoot apical meristem the plant cells divide and begin to differentiate into different cell types (such as different cells of the leaf, or vascular cells). Later growth (of, say, a leaf) is largely a result of cell expansion (although cell division does still occur, but drops off as the leaf expands). Therefore, if you punch a hole in a leaf, it probably won't be filled in because the cells in that leaf have finished growing and dividing.

However, as a shoot grows, more meristems are created. These are found in the axillary buds, just above where the leaf meets the stem. The meristems in the axillary buds can grow to form branches. Different plants obviously make different numbers of branches, but there is a common control mechanism known as apical dominance, where the meristem at the tip of the shoot suppresses the growth of the lower axillary buds. This is why a shoot with no branches can be made to grow branches by cutting off the tip (gardeners often do this to make "leggy" plants more bushy).

All of that was a long explanation to say, no, a plant doesn't normally^^ regenerate in the sense of filling in cells that have gone missing. However, if you cut off a shoot, the next remaining bud might begin to grow and, in a sense, replace the part that was lost. In that case, an existing bud is recruited to form a new branch and replace lost functionality, but I wouldn't say that qualifies as regenerating missing tissue.

^There are other types of meristem as well.

^^If you torture plant cells enough you can force them to become "stem cells" and thereby make an entirely new plant, but this is rare in nature.