NO, he does not NEED to eat. He CAN simply die and resurrect.
However, the mechanics for that state do not explain whether he suffers, starves to death over time and dies or whether he is returned to his state of temporal grace, fit and able to starve to death over a period of thirty days, again.
What we have observed
Jack Harkness cannot die. Or perhaps we should redefine this as "he cannot stay dead". Any condition that renders him dead is invalidated and he returns to life. Starving to death certainly qualifies. His condition is due to an improper use of the Temporal Vortex as applied by Rose Tyler.
Jack has been shown to eat. He appears to relish it, in fact. Despite his "temporal immortality" he appears to need to live as a normal human in all other ways. He has not been shown to be invulnerable to environmental conditions, he has been shown to get tired and become fatigued. So it stands to reason he would get hungry and potentially die from starvation.
It would be safe to assume Captain Jack Harkness needs to eat to continue living. Without food he will certainly expire, however, the mechanics which describe his "immortality" are tenuously described, at best.
Dying is Easy
We have watched Jack Harkness die in any number of exotic ways but we have never seen him starve to death. Curious. Why not? Can he starve to death, what would it look like? More importantly what does his resurrection look like?
During one of the most terrible scenes, the good captain is encased in concrete to prevent his escape. This means he is denied air, water, and food. When he is rescued, he appears none the worse for the wear, physically. But such an imprisonment did not allow him to starve...
We are never allowed to see him return from his emaciated body to his robust vigorous self.
But we have seen him tortured and flesh stripped from his body, only to watch it reverse and be restored back to his soon-to-be-living form.
Since the mechanic for his dying and resurrection are unclear:
it is not known whether he is returning to a singular spot in time (pre-starvation)
or whether he is being resurrected in his starved state only to die again possibly in minutes.
Does this mean he starves to death again in a few hours or will he return and reset to a point where he is physically robust and then die in the amount of time he would have if he started healthy and then found himself without food or water?
Realize the human longevity without air food or water is: three to five minutes without air, without water he would last about three to five days, without food perhaps twenty to thirty days, assuming water was available in sufficient supply.
Our current experience says he returns to life, UNDAMAGED, indicating a possible return to a previous temporal state. But no matter his condition, he will resurrect, starved or not, and likely continue until he expires again. Judging from his nearly destroyed body event, I tend to go with him returning from a state of temporal grace, no matter how he dies.
Temporal Grace
If we assume he is restored to the state he was in when he died how do we explain:
His retention of memories up to and including whatever he knew before he died? If he were physically being restored from a temporal backup, he would have no memories up to the point of his death.
It is more likely, any damage which manages to kill him is undone and he is restored to a point just before he died. Since this is more likely, he will return to the state prior to his condition leading to his death.
Without the specific chronal mechanic for his resurrection, WE CANNOT KNOW FOR CERTAIN. Neither Torchwood nor Dr. Who have supplied us with sufficient information to make that determination with any degree of accuracy.