How long before lidocaine side effects wear off?

Thank you for your question. 
What you are experiencing is not normal from lidocaine or lip filler injections. The sudden dizziness and almost passing out could be what’s called a vasovagal response and could be a normal response from pain of the injection. 
But the symptoms you have described 24 hours later in your arm and leg are doubtfully related to your lip filler.
Going to the ER was an excellent choice.  Please follow up with your primary care Dr and/ or a neurologist to help determine the reason for these symptoms 

Whatever your symptoms are, are not related to injection with the lip filler or the lidocaine. You should be evaluated buy a neurologist. Good luck from New York City

A reaction to Lidocaine are very rare, most of the time it is reaction to paragons, the preservatives used in multi-dose vial. In general the non-immediate reactions are reactions that may occur with delay after the offending agent was introduces. However, the symptoms you have described do not match what is known about delayed reactions. Your symptoms seem to be clearly of a systemic nature involving your entire body; the descriptions is very similar to those of hyperventilation.  I hope you have received explanation during your ER visit. Good luck!