It seems to me that it does not make sense to include my teaching statement in an application package for a research associate position (UK), which is focused on a specific research project (the only teaching-related activities there might be presentations, seminars or similar events). Could including teaching statement hurt or, vice versa, benefit the application and subsequent decisions?
I wouldn't hold it against anyone personally, but it could be bad or it could be good. If there are teaching or training opportunities as part of the job (like there are for people I hire), then it might be marginally helpful. It at least indicates an interest in the ancillary functions of the role. However, if it's a pure research position unattached to an organization with no possibility of a teaching role, a particularly unkind or paranoid sort might hold it against you since it could serve, to them, as an indication you'd prefer a professor/lecturer position, and that you are likely to not stick around very long if you can find one. This kind of stuff matters a lot more to some folks than it does to me personally. There's some legitimacy to this kind of thinking since the hiring process is time-consuming and expensive. You don't want to lose someone shortly after getting them on board.
My recommendation is to stick to the requested materials and no more. Tailor your application, CV, and letters to highlight the things that the position description wants.