Here's a big one with the pro-Open Borders crowd. They claim that the European colonists who founded the United States were illegal aliens. It's based on the flawed assumption that the Native American tribes had similar laws governing immigration that we do.
The reality is that prior to the appearance of the Europeans, the Native Americans had no real immigration issues to deal with and so no real laws governing them. They weren't ocean-going so they had no real concept of what lay over the horizon; how many people were there, what their culture was like, or what they might do if they gained a foothold in the Americas.
When the Europeans reached the Americas they were frequently welcomed by the natives. Sometimes this was because the Europeans brought them presents, trade goods they couldn't provide for themselves. Sometimes it was because the Europeans offered to be valuable allies in wars with neighboring tribes. The Europeans weren't illegal aliens because the native tribes largely had open border policies and those policies ultimately lead to their being forced out of their homes and confined to what lands the European immigrants deigned to give them.
The founding of the United States actually provides a perfect example of why open borders are a bad idea. If a nation fails to control its borders it will ultimately be overrun by people who do not have the nation's best interests at heart.