No. Based on what people post about drug testing at Wendy's, the company, as it appears, doesn't require that as part of the application procedure.
On the other hand, Wendy's Employee Handbook states (under its Alcohol and Drug Abuse Policy) that:
Wendy's may, without notice, require an employee to undergo testing for evidence of the use of alcohol, illegal drugs or other substances if, in the opinion of Wendy's management, there is reasonable suspicion that the employee is under the "influence." It is a condition to continued employment for an employee to submit to substance abuse testing (i) when the employee is involved in an accident, (ii) as part of a follow-up program for drug or alcohol abuse, and (iii) when there is reasonable suspicion to believe that the employee is illegally using drugs or abusing alcohol.
The above statement is clearly being addressed to employees who are working for the company already.
To search for available positions at Wendy's near you, go to careers.wendys.com.
Check out Job Interview Tips for Fast Food Restaurants from Chron.com.