I haven't been able to find out what the situation in the States is. But in Germany, for instance, it is definitely illegal as it is considered document forgery. It is thus punishable by up to 5 years in prison, or even 10 years in severe cases[0].
In fact, AFAIK a company is not even allowed to issue an invoice twice without clearly labeling it as "copy of the original" and most companies must keep the invoices they issued as well as their books (and thus the invoices they received) for at least 10 years[1].
I suppose the legal situation must be very similar in other EU countries. One reason being, for instance, that invoices are used (by companies that deduct VAT) to request back from the tax authorities the VAT that one pays on invoices by other companies (in the EU). The invoice you receive from a company must therefore match that company's books. All hell would break lose if everyone could just forge invoices, whether with bad intentions or not.
Case in point: Here[2] is a guy asking for legal advice as he's being charged with document forgery even though we wasn't trying to defraud anyone or anything. Specifically, he had lost the invoice of the TV he was trying to sell on Ebay, so he ended up forging it because Ebay required people to upload the original invoice to sell TVs on the marketplace.
Invoices, and more generally the receipts underlying bookkeeping transactions, are the fundamental building block of reliable bookkeeping (and company audits, mind you).
