As someone who believes in the inerrancy of scripture, I must admit that this particular question has baffled me more than any other objection I have faced. After all, if the Bible is God's word, then it must be true. However, if one were to pick up an English Bible today and begin reading through the Old Testament one would quickly find that the Bible—which is supposed to be the infallible Word of God—plainly states that bats are birds! Clearly this is not true. Bats are mammals. So how can one reconcile such an obvious scientific blunder in God's Holy Book? If God's Word is true then how can it promote something so blatantly false?
As it turns out, this conflict between the Bible and modern biology does not stem from a lack of scientific knowledge on the part of the Old Testament. Rather it is an issue of translation from Hebrew into English.
The word translated as bird in English Bibles is the Hebrew word 'owph, (עוֹף) which is in fact a much broader term that encompass not only birds, but all winged creatures including insects and bats. Therefore, a more accurate translation would be a creature with wings or a flying animal, not bird. However, this word is most often used in reference to feathered birds, and in the particular case of Leviticus 11:13-19 the majority of the creatures mentioned are in fact birds. The only exception being bats, which are mentioned last. For this reason, English translators chose to render 'owph as bird in English Bibles, though the more liberal usage of the word as it pertains to all winged creatures would be far more accurate.
That said, some have argued that the grouping of bats with birds in this passage is completely consistent with our modern methods of scientific classification. Leviticus 11:13-19 as a whole describes members of the Phylum Chordate, which includes birds and mammals. While Leviticus 11:20-23 identifies members of the Phylum Arthropoda. Verses 13-19 groups feathered birds together within Class Reptilia, which is in complete agreement with modern scientific opinion. Finally, the last part of verse 19 places bats in Class Mammalia by themselves, which is again consistent with modern taxonomy.
In the case of Deuteronomy 14:11, the word used to describe clean birds is tsippowr (צִפּוֹר) which specifically refers to sparrows or other small birds that skip about on the ground. This is further clarified by the change in grammar found in Deuteronomy 14:12, which separates the behavioral characteristics of sparrows from the traits of the other flying animals and insects listed in verses 12-20. Furthermore, neither the word tsippowr or 'owph appear in the Hebrew version of Deuteronomy 14:12. This means that the word bird or birds found in some translations is an addition made for the sake of English grammar, and is not part of the original text.
It is also important to note that in Deuteronomy 14:20 the author returns to the word 'owph to describe clean animals and insects, again testifying to the separate meaning of the specific word tsippowr as it relates to the more general term 'owph in ancient Hebrew.
That being said, it is important to view this question in the proper context. In both passages it is not God's intention to classify animals into their modern taxonomic Phylum and Genus. Instead, God is giving the people of Israel clear instructions on what kinds of flying animals symbolically represent life, purity and holiness (clean) and which ones symbolize death and impurity and should be avoided (unclean).
In short, the Bible is not in error when it places bats among the flying animals that are unsafe to eat. God did not mistakenly identify bats as birds when He gave his Law to the Children of Israel. The confusion arose thousands of years later as a result of translation into a foreign language, and should not cause anyone to lose faith in the inerrant Word of God.