In the ORIGINAL comics, it wasn't a lasso of truth at all! Truth was just a side effect. In the early comics, the lasso compelled OBEDIENCE on anyone who was bound by it.
Obviously, if Wonder Woman wanted someone to answer her question, she would simply use that property. If she says "Answer truthfully" as a command, then not speaking is not an option, as we see in this case where you can imagine the person (a foreign national on a mission for their country) might choose silence rather than truth, if that were an option.
But that wasn't its only use, even into the 70s (though this particular scene is from '67) it was used for non-truth related control:
(images taken from this CBR article: When Did Wonder Woman's Golden Lasso Become a Weapon of TRUTH? which also contains a much more comprehensive history of the changes over time)
It was only post-crisis that it explicitly became a lasso of truth, but even then it had other powers. The DC wiki describes this version's powers as follows:
The Lasso also appears to be able to alter in length, based on it's wielder's needs. It is able to restore people's lost memories, hypnotize people, get rid of illusions, and people who are in its circumference will be protected from magical attacks. Empowered by the Fires of Hestia, the Lasso forces anyone held by it to tell and understand the absolute truth. Recently, the Lasso has been shown to actually transport Diana's mind into that of whomever is tied within it.
By this description, it seems that no they cannot refuse to answer.