Yes, at least in Tennessee. On December 21, 2015, the Supreme Court of Tennessee amended its bar admissions rules to include this paragraph:
(b) The Board or any individual member thereof, as part of the character investigation of an applicant, may request an applicant to submit to a drug test. Failure or refusal to submit to the drug test shall be sufficient cause for the Board to refuse such applicant a license.
See Order, In re Amendment of Rules 6, 7, and 8 (RPC 5.5), Rules of the Tennessee Supreme Court, No. ADMIN2015-00443 at p. 16 (filed Dec. 21, 2015).
Louisiana does not have an express rule permitting the drug testing of bar applicants. However, it is not uncommon for the Committee on Bar Admissions to require an applicant with a history of drug or alcohol abuse to submit to an evaluation by a mental health professional approved by the Louisiana Judges’ and Lawyers’ Assistance Program (JLAP). Such evaluations could include a drug testing component. Moreover, such an applicant could be admitted by the court conditioned on a JLAP contract, which typically includes random drug testing. See La. Sup. Ct. Rule XVII § 5(M).