In D&D 5th Edition, if you use the Beast Sense spell on a creature with truesight, do you see as if you had truesight?
It doesn't matter; there are no beasts with truesight
The spell beast sense (PHB, p. 217) says:
You touch a willing beast. For the duration of the spell, you can use your action to see through the beast’s eyes and hear what it hears, and continue to do so until you use your action to return to your normal senses.
As you can see, the spell specifies that you cast the spell on a beast, meaning a creature with the beast type.
As it stands, there are no officially released creature of type beast with truesight.
But what if there were?
If, let's say, your DM had homebrewed a creature that was a beast that did have truesight; this would work in the same way as a beast with darkvision.
Since you "see through the beast's eyes", you would gain the benefits of its senses, since you're using its senses to see. So, in answer to your question, yes, you would then effectively get to use their truesight whilst seeing though their eyes.
My take on this matches up with the consensus of this related question: Do I still retain my own vision type while seeing through my familiar's eyes? (thanks to Medix2 for finding that question, see the comment under the question).