Is teaching the Bible in schools illegal?

It depends. Teaching about The Bible such as in a comparative religion or comparative literature course is perfectly legal. However teaching the Bible itself is illegal per Abington Township School District v. Schempp (consolidated with Murray v. Curlett), 374 U.S. 203 (1963) which established a three part litmus test: sponsorship, financial support, and active involvement of the government in religious activity.