Crossing the US-Canada border by car through visitor visa

Canadian (or US) border agents do not usually check your driver's licence (unless you are travelling on an EDL as an American/Canadian citizen).

However, driving on an expired licence may be illegal in the U.S. for you. Police checks can happen independently to immigration and customs checks, and your insurance coverage may be invalidated.

In Canada, if you are a temporary visitor who has only been in the country for a short time, you are usually able to use a valid foreign licence to drive a standard car (from U.S. or India, as long as it is not expired) (but provincial rules differ in details).

Immigration-wise, you need a valid Canadian visa and qualify for one of the exceptions to the current travel restrictions.

Only U.S. citizens and permanent residents are eligible for the fully vaccinated exception to the current Canadian border restrictions. From September 7, 2021 (subject to change, of course), the Canadian government intends to allow all fully vaccinated foreign nationals to enter for discretionary purposes. If you do not qualify for such exceptions, you will not be allowed entry into Canada.

If you plan to drive back to the U.S., note that a border restriction remains in place for Canada-US land borders until at earliest September 21, 2021. If you do not qualify for "essential travel", e.g. as a visitor, your entry could be denied. If you leave U.S. before your status change is approved, your application will be cancelled and you may not be able to return to the U.S. without a new visa if you have no valid status.

You have two separate concerns:

1. Immigration. Canada will want to see a valid passport and Canadian visa. They don't care about your driving license. Likewise the US wants to see you're eligible for re-entry on the way back, which in turn depends on your current I-94 status and other documents, but not the driving license. If you want an answer related to whether or not US immigration will allow you to come back, please ask a separate question on Expats.SE. Source: I drove around 15 times across the Canadian border.
2. Driving. Its illegal to operate a motor vehicle without a valid license. This has to be either a valid local license or a foreign valid license. If you don't have either, you're not allowed to drive. Even if this is not being enforced by the border authorities, you could still be stopped by the police in either country.