In 1 Corinthians 8:4,7, Paul says:
Therefore concerning the eating of things offered to idols, we know that an idol is nothing in the world, and that there is no other God but one.
However, there is not in everyone that knowledge; for some, with consciousness of the idol, until now eat it as a thing offered to an idol; and their conscience, being weak, is defiled.
Idols and what they represent are nothing but pagan superstition, so offering meat to an idol does not change the meat in any way.
But if someone still has underlying doubts in their faith, then such meat will have a mental effect on that person.
If they eat it with even the slightest thought about its association with a pagan god, they are implicitly acknowledging the power of that god.
Paul is telling people that if they are with recent converts or other people whose faith isn't as strong as theirs, they should refrain from eating such food, as it might encourage those others to eat it too, at a time when they are not yet ready to eat it without experiencing pagan associations.
(In Polanski's movie, when someone tries to frighten a vampire using their crucifix, he responds: "Oy vey, have you got the wrong vampire."
To the scriptwriter, who did not believe in Catholicism, the power isn't in the object itself, but in the believer's mind. Paul is expressing the same idea with respect to not believing in idols or the non-existent gods they represent.)
In Mark 7:19, the parenthetical statement, "(In saying this, Jesus declared all foods clean.)" appears in only some translations, and doesn't appear at all in the original Greek text.
Had Jesus said such a thing, the listening Pharisees would hardly have stayed quiet about it, but there is no record of any such reaction.
Had Jesus said such a thing, the Disciples wouldn't have continued to refrain from unclean food, as Peter claimed in Acts 10, years after the Crucifixion.
This added comment should be ignored.
Acts 15:20 is more relevant to this question.
The Christian leadership was debating whether Gentiles had to convert to Judaism before becoming Christians.
The decision was that no, much of Judaism was part of Israel's covenant with God, and as such it should not be required of converts.
However, Jews did (and still do) believe that God gave seven laws to all mankind at the time of Noah.
Four of them are generally accepted by almost all religions: blasphemy, murder, theft, and injustice, and so they could reasonably expect any Gentile converts to already follow them.
The other three laws are less ubiquitous: worshipping idols, sexual immorality, and eating meat that has not bled to death.
It is these three laws that the leadership decided must be followed by Gentiles while they are converting to Christianity.
That is the origin of the three conditions mentioned in Acts 15:20.
For more details, see Were there implicit laws not referenced in the Acts 15 letter to gentile believers? - Christianity Stack Exchange
So yes, Christians are expected to refrain from eating meat that hasn't been drained of blood before the animal's death.