It is more complicated than that. But yes, mitigation is essential. Even if the tenant doesn't move in, the tenancy commences on the day that the tenant is entitled to occupy the rental unit. So the tenancy starts, like it or not. Rent continues to be owed until the date that rent would have been owed to had the tenant given proper notice of termination. If there is no term (month to month), that would be 60 days minimum effective the last day of the lease term. So if today you said "I'm leaving on April 30th", you would have an obligation to pay rent to May 31st, subject to mitigation efforts by the landlord. Notice must be given on a proper LTB approved N9 notices. No proper signed notice...no obligation to mitigate.
Now what if there was a one year term. Tenant moved in December 1st on a one year term. Tenant then says (or says today, March 5) I'm not moving in (or I'm moving out April 30th). Same rule. Rent would continue to be owing until the earliest date that it would have been permissible to end the tenancy by notice, that is, December 31st. However, a court is required to consider whether the landlord made efforts to re-rent when determining what rent is owing. So no mitigation, the Court will likely say "no rent owed". And again, in lease term, must use a propr N9 notice.
If a landlord and tenant AGREE to end the tenancy (tenant gave 47 days notice, landlord knows he can re-rent quickly, it's a hot building), then they can sign an LTB N11 form. If they sign an N11 form, the date they agree to (even if it's next Tuesday) will be the last date to which rent is owed, assuming the tenant moves out.
If the tenant signs an N9, or if the landlord and tenant jointly sign an N11, and the tenant doesn't move out, or the landlord wants to make sure he/she will move out, the landlord can file an ex parte application, an application without notice to the tenant and without a hearing. The landlord has to include a copy of the N9 and N11, has to provide an affidavit swearing to the circumstances around the notice or agreement, and must file the application (called an L3, with a $170 application filing fee) within 30 days of the termination date in the notice or agreement. The LTB will order Sheriff eviction based on the L3 application.
Hope that helps. See s.37, 43 and 88 of the Residential Tenancies Act for specifics, or call the LTB hotline at 416-645-8080.