The mishna in Tractate Shabbat[1] assumes that one may feed animals on Shabbat and discusses which systems are permitted and which are not. The gemara[2] presents opposing sources that indicate that it is forbidden to feed animals in any manner, and then provides two distinctions to resolve the apparent discrepancy. The main difference is that one can feed only those animals whose food is his responsibility. The gemara also distinguishes between those animals that are dependent on man for food and those that can find food independently. The Shulchan Aruch[3] accepts the stringency of both distinctions and says that one is permitted to feed an animal only if he is responsible for it and it needs to be fed. (When the owner may feed, others may do so on his behalf.[4]) Otherwise, it is in the forbidden category of unwarranted toil on matters not related to Shabbat.[5] Regarding a rabbinic prohibition of this type, it is unusual to make an exception for those who enjoy feeding animals.
The main determining factor of whether one has a responsibility to feed animals is whether he owns/controls them. If he takes them to his home, barn, etc. for his benefit, then he has a serious responsibility to provide for them properly. Apparently, members of the animal kingdom that are not in the human domain are in the Divine domain, and HaShem looks out for their needs. Thus, it is forbidden to fill a bird feeder, which attracts and provides for wild birds, on Shabbat.
Some poskim allow, under certain circumstances, feeding an animal that one does not own. The Shulchan Aruch[6] mentions that one may feed a dog. Whereas some explain that this refers to one’s own dog, the Magen Avraham[7] says that it refers to any dog and that it is a special dispensation based on the gemara[8] that mentions HaShem’s concern for dogs’ difficulty finding food. Some prominent poskim[9] extend this concept to any animal that we know is hungry and suffering. The same might apply to birds during the time of the year when one knows they have difficulty finding food, especially if he accustomed them to frequenting the area near his home and the surrounding area does not have sufficient food opportunities. (We cannot offer a zoological opinion, and certainly not about circumstances we have not observed.)
Since the issue of feeding on Shabbat is unwarranted toil, many say that if one needs to shake out his tablecloth for his own purposes, he may do so intentionally in a place where animals will benefit.[10] An interesting, longstanding machloket is whether the minhag of some people to throw breadcrumbs to birds on Shabbat Shira (to recognize the birds’ assistance in glorifying the miracle of the manna that fell before Shabbat) is permissible. The Magen Avraham[11] and Mishna Berura[12] object because one is feeding birds that he does not own. The Aruch HaShulchan[13] and others justify the minhag by claiming that we throw the crumbs for our sake, not theirs.
On Yom Tov, it is permissible to slaughter animals in order to eat their meat that day but there are several restrictions, including that one may not trap them on Yom Tov. In order to distance us from the possibility of taking animals that we should not, the Rabbis forbade throwing food on Yom Tov to animals that are off limits to our use.[14] In such cases, Yom Tov is more stringent than Shabbat.
In general, the logical suggestion is to fill the bird feeder before Shabbat (or Yom Tov). If he failed to do so, he should not, under normal circumstances, put out food for wild birds on Shabbat; he can assume that the birds’ Maker will provide for their welfare.