As the question states, can a really dirty or clogged air filter cause engine to burn oil in anyway?
Yes, it could possibly.
A really dirty air filter will cause a greater restriction in the intake tract. This will cause greater vacuum levels. If your engine has weak/worn oil seals (such as valve seals), having higher vacuum can pull oil past these and cause the engine to burn oil.
Brought to my attention (in the comments), the higher vacuum could also exacerbate oil being pushed past the oil scraper rings if they are worn. Also, depending on how the PCV valve is situated/engineered, you might see excess oil getting pulled past it as well.
Restricted air flow will cause a richer fuel/air ratio and result in a black-ish smoke emitted from the exhaust and a black sooty coating at your tailpipe. A richer mixture may cause a wash down of the layer of oil in your cylinders and therefore the rings might lose their ability to keep the crankcase oil down where it belongs. If you're sucking oil into the combustion chambers the resulting burning oil will cause a blue-ish grey smoke. You mention nothing about the condition of your exhaust gases.