I'm going to try to keep this brief.
People say that in order to preserve the presumption of innocence in rape trials, accused rapists cannot (must not) bear the burden of proving that the sex between accused rapist and survivor was consensual.
This is (or should be) wrong.
So what is the presumption of innocence anyway? The presumption of innocence is a legal principle that guides the trier of fact (the judge or jury) when they weigh the evidence to arrive at a verdict. It tells the trier of fact that the accused is deemed innocent until the prosecution advances sufficient evidence to prove (beyond a reasonable doubt) that the accused committed the crime in question. In other words, the presumption of innocence challenges the prosecution to "prove it or lose it".
What the presumption of innocence does not require from the prosecution is evidence to refute an affirmative defence. For example, in order to preserve the presumption of innocence in a murder trial, we
some jurisdictions* do not require the prosecution to proactively refute the claim that the accused was defending herself.
What does this have to do with rape? Well, conventional wisdom is as I described above. The prosecution must prove that sex did happen between the two specific people, and that the complainant did not consent. The way laws are set up, this is true. However, it need not be the case, and the fact that it is the case shows that we have some peculiar ideas when it comes to consent for sex.
I want you to think about theft and property law for a moment. You own things. The fact that you own them means that other people can't just take them without your permission. And we presume that, absent evidence to the contrary, you don't permit any random person to take off with your things. If someone takes something that belongs to you, the law does not require you to prove that you withheld permission. If the person who took your property wants to claim that you gave them permission--in other words, that you consented--that is on them. This does not violate the presumption of innocence, because the prosecution always has to prove that the person who took your property actually took your property.
For some reason, we don't see consent to sex in the same way. When we expect the prosecution in a rape case to prove a lack of consent, what we are saying is that the default position is one of consent. That strikes me as very weird and wrong. In order to keep things consistent, we should take the same approach to the right to sexual agency and bodily sovereignty as we do to the right to ownership and exclusionary control of property. We ought to presume that people default to a state of control and exclusion unless they give some indication to the contrary. This will not violate the presumption of innocence in rape cases, given that the prosecution will always need to prove that the accused had sex with the alleged survivor (and this is by no means trivial). I think going this way is better than applying rape-reasoning to theft cases, for what should be obvious reasons.
* Credit to mcdermott