Does it matter if your decision might not matter? I have read several experimental papers which ask people to make decisions which only matter with a certain probability. For example, individuals might be asked to submit bids in an auction, but told that their bids only 'count' with a certain probability $p$ (with probability $1 - p$, their payoff is 0 regardless of what they do). Alternatively, if we wanted to learn whether an individual's valuation for a good exceeds £10, we could offer to sell her that good for £10; but tell her that the good will be sold to her (and the payment taken) only with a certain probability. In both cases, the idea is that individuals should act as they would if $p = 1$ (i.e. the mechanism is 'incentive compatible') which then allows researchers to run these experiments more cheaply than they would be able to otherwise.

Is there any evidence about the general validity of this approach? More concretely, is there evidence as to whether the outputs of such experiments vary with $p$? Intuitively, I would have expected that subjects become inattentive if $p$ is very low (since in absolute terms, it doesn't matter much what they do), which could then add noise to the elicited estimates.
Interesting question. I don't have any empirical evidence to offer, so I can't answer your question if that's what you're asking for.

But it's easy to provide "theoretical evidence", as it were, that your concerns may be well-founded, even without leaving the EU model.

You simply need to add a decision cost --- or a cost to figure out your true WTP --- to the decision-theoretic model of your participants. 
Then, as the EU stakes decrease (because $p$ decreases), participants are less likely to invest in the introspection cost of discovering their true WTP.
As a consequence, participants' behavior become a much noisier indication of their WTP: 


Some of your participants will decide to buy the good although their true WTP is lower than the price. This could happen if the participant decided not to invest in introspection and therefore got a sufficiently upward-biased signal of their true WTP. 
Other participants will decide not to buy the good although their true WTP is above  the price. This could happen if the participant decided not to invest in introspection and as a consequence got a sufficiently downward-biased signal of their true WTP. 


Assuming the cost of introspection does not change with $p$, participants become less likely to invest in introspection as $p$ decreases (because decreasing $p$ decreases the stakes, and therefore the benefits of introspection), and you'll get more of this noise as $p$ decreases.