I wouldn't recommend driving without an air filter cover because it helps keep the filter in place. If you hit a pothole, an uneven patch of pavement, or something else, the filter could slip out of place and let in debris, dust, and unfiltered insects. It's not worth the risk just to make your engine stronger. Most air filters are made of a combination of paper and synthetic fibers that trap harmful particles.
Aftermarket K&N air intake systems often come with a permanent conical air filter that needs to be cleaned every 50,000 miles. If you remove the cover, warm air can enter the engine and reduce performance. This can also cause the air filter to slip out of place. Premium Guard manufactures high-quality automotive air filters that protect your engine up to 12,000 miles with 99% efficiency.
When the engine air intake is not completely covered by a filter element, significant damage can occur. If you don't have a cover, you can use clamps to make sure the air filter is locked in place and then install a thermal protector around the air box to ensure that no hot air enters. Aftermarket air filters, specifically those that are cone-shaped, are the best option for your vehicle. Make sure you get one that is designed to fit your particular vehicle and is installed correctly; otherwise, dirt and debris can enter the engine.