Word for someone who propounds a point despite all basic rational evidence against it
Perhaps a propagandist!
From MW dictionary:
Simple Definition of propaganda : ideas or statements that are often false or exaggerated and that are spread in order to help a cause, a political leader, a government, etc.
propagandist: noun or adjective
More from Wikipedia:
"Propaganda" is information, especially of a biased or misleading nature, used to promote or publicise a particular political cause or point of view. Propaganda is often associated with the psychological mechanisms of influencing and altering the attitude of a population toward a specific cause, position or political agenda in an effort to form a consensus to a standard set of belief patterns.1
Propaganda is information that is not impartial and is used primarily to influence an audience and further an agenda, often by presenting facts selectively (perhaps lying by omission) to encourage a particular synthesis, or using loaded messages to produce an emotional rather than a rational response to the information presented.
Propagandists frequently argue their point by claiming that the other side is attempting to take away your control. For example, Republicans frequently claim that Democrats are attempting to control you by imposing big government on your private life and take away your spending power by imposing higher taxes while Democrats frequently argue that they are reining in big corporations that are attempting to influence elections with money, power and take away your job, health etc. ... According to bipartisan analysis, these claims are frequently untrue.
I would call such a person a mythomaniac:
American Heritage dictionary
[a person who has] a compulsion to embroider the truth, engage in exaggeration, or tell lies
Or for the particular example that you gave:
mudslinger
Merriam-Webster
: one that makes malicious attacks especially against a political opponent
pseudologist
dictionary.com
- lying considered as an art.
Making things up, lying, misdirecting, mistakenly attributing, and bullshitting are variously used in different situations for actions such as these. The person who does them can be called confused (very charitable), mistaken, a liar, etc. We will likely see all of these a lot more during the 2016 U.S. election season. (Here is an article ruminating on just why this particular tweet was sent.)
You might use something like "tunnel vision" for a slightly more sincere person - it implies that they keep repeating the same point because they can't see or don't believe evidence against their pet theory, rather than because they know it's false and want to convince people otherwise.
Tunnel vision, as a metaphor, denotes the reluctance to consider alternatives to one's preferred line of thought; this could include physicians treating afflictions, detectives considering crime suspects, or anyone predisposed to a favored outcome. - quoted from wikipedia.org
Such person might "have blinders on" (implying the stance they took is preventing them from looking around for other evidence). Or they might "be blinkered", which comes from the same source - originally screens used to keep horses from looking around or being spooked by other happenings nearby.
To have blinders on - to not be able to recognize or accept what is happening around you. Etymology: based on the literal meaning of blinders (pieces of material attached next to a horse's eyes to keep it from seeing anything on either side) - quoted from idioms.thefreedictionary.com
blinkered (ˈblɪŋkəd) 1. considering only a narrow point of view 2. (Horse Training, Riding & Manège) (of a horse) wearing blinkers
This person might also be described as hyper-focused on their point, or irrational, or obsessed. Any terms like that.
I realize this is a more generous interpretation of such people than your request implies - since these terms assume that the person is wrong, but might genuinely believe their point. However, I think actual evidence against something is not necessarily a barrier to people believing it
Using delusional or deluded might be appropriate.
The issue specific to Donald Trump is it’s almost impossible to understand any of the motivation to his behavior. For all any of us knows, he really believes what says and does is true; that is what is really scary about him.