Obama and Biden both have had to deal with irrational conspiracy theories. Neither conspiracy theory stands up to scrutiny or survives a logical look.
Obama had the Birthers' Conspiracy Theory
The Birther Movement falsely asserted Obama was ineligible to be President of the United States because he was not a natural-born citizen of the United States as required by Article Two of the Constitution. Birther conspiracy theories were predominantly held by conservatives and Republicans, as well as racists.
Despite the fact that Hawaii's Department of Health confirmed that the state held Obama's "original birth certificate on record in accordance with state policies and procedures", Birthers were skeptical.
The Birthers' theory was that Obama was really born in Kenya or someplace outside the United States. Birthers believed that Obama's parents, in 1960, realized that the infant boy Obama might be a candidate for president someday so they had fake US birth certificate made up. Birthers don't offer an explanation as to why his parents thought a biracial kid wouldn't be hampered politically by having distinctly foreign sounding name.
Nor do Birthers answer the question if Obama was born elsewhere, how did he arrive in Hawaii? He would have either come on a ship or a plane, yet there is no passenger manifest with his name or the name of his parent(s) coming to Hawaii. Records are kept on that in particular for immigration purposes. People can find the names of ancestors who came by ship and disembarked at Ellis Island over 100 years ago.
Biden has the Dementers' Conspiracy Theory
The Dementers' theory about Biden is that his brain is shot with Alzheimer’s disease or some other type of senile dementia. You might wonder why a person with dementia would want to run for president and more miraculously win the presidency. (Even those who believe the last election was stolen must admit that Biden collected many millions of votes.) They say that Biden is the puppet of nefarious, unseen, elder abusing handlers. Somehow these handlers juice Biden up so he can perform adequately from the podium including in debates. The Dementers believe that these handlers are secretly manipulating Biden to impose a liberal agenda upon America.
Presumably Biden's handlers are political savants who were able to take a less-than-ideal candidate and win the presidency. It doesn't take Mr. Spock to ask the logical question, if they were so smart why didn't these handlers select a younger, better looking, more oratorically-gifted presidential candidate?
Also, who are these handlers? How have they managed to exist in the shadows while the news media maintains a reportorial spotlight on the presidential administration at all times?