I was told something would happen at Tate Modern on Monday, but not exactly what. In the event, had you wandered through the London museum mid-afternoon, you would probably have missed the three activists winding their way through the galleries, whispering chunks of the transcript of the BP Deepwater Horizon trial into phone-sized cameras.

Last year, Liberate Tate, the group founded three years ago with the aim of ending the sponsorship of Tate by BP, delivered a wind turbine blade to the gallery. The year before, a naked man covered in oil curled up in a foetal position on the floor. By contrast this week's "performance", timed to coincide with the third anniversary of the notorious oil spill, is low-key.

The signs are that the museum will let it go ahead as planned, for an hour each afternoon this week, in the building and live-streamed online. "We obviously can't comment on how we will respond to events which have not happened yet," said a statement, "but the events on Monday did not break any gallery rules." A BP spokesman said the company is "very proud of its partnership with Tate, which spans more than two decades" and "of course we respect the right of people to express their views".

The museum's 5 million yearly visitors have no way of knowing it – BP appears fourth on a list of supporters on the wall – but the relationship between the oil giant and the museum is a special one. BP is the museum's longest-standing sponsor, and when Tate Modern celebrated its 10th anniversary in 2010 the party also honoured 20 years of sponsorship of the Tate galleries by BP.

The day after BP chief executive John Browne, now Lord Browne, left his job following the revelation that he had lied in court about the circumstances in which he met a former boyfriend, Tate director Nicholas Serota wrote an article that claimed BP's support for the arts was "the catalyst" for a huge upsurge of interest and that "we should all be grateful". Three months after leaving BP, Browne became a Tate trustee. In 2009, he was promoted to chairman, a position he will hold until 2015.

This week’s 'performance' is timed to coincide with the third anniversary of the Deepwater Hoprizone oil spill. Photograph: Graeme Robertson for the Guardian

While Lord Browne is no longer formally linked to BP, a freedom of information request submitted on behalf of the activist group Platform, with the aim of finding out much more about the Tate-BP relationship, argues that his ongoing energy interests (he is director of fracking company Cuadrilla and sits on several boards), made him an "inappropriate" person to reject the FOI request on Tate's behalf.

Campaigners now await a ruling by the information commissioner and say they are confident that the public interest in the sponsorship of a national institution should outweigh the commercial considerations that the commissioner is obliged to balance against this. If their appeal is successful it would be the first time the information commissioner has forced the disclosure of sponsorship details.

In the museum on Monday it was not easy to tell what degree of public support such a move might command. Funding for the arts is always vulnerable when public spending comes under pressure, and philanthropy and sponsorship are widely accepted as part of the package, with museum and theatre-goers habituated to lists of tiered supporters and "friends". None of the visitors I approached at Tate Modern thought BP's environmental and safety record, or the role of fossil fuels in accelerating global warming, should place the company beyond the pale of corporate sponsorship, along with tobacco and arms companies and foreign dictators.

But nor did anyone want to criticise the protesters. "It's a place of arts. Should politics should be in a place of arts?" said Nicolas Durand, 27, a teacher leading a group of French students. "I don't know, that's why I put it as a question."

Liberate Tate has received some support from more established figures including painter Raoul Martinez. Theatre director Katie Mitchell, who last year collaborated with scientist Stephen Emmott on a monologue about the environment, says the arts community "musn't get in a situation where our desperation for money means we're turning a blind eye to who we're taking it from, or to what they're doing. I don't know if it's possible for an oil company to have an environmental programme that is credible."

But the more typical reaction is ambivalence. Turner prizewinner and Tate trustee Bob and Roberta Smith says he opposes deep-sea drilling but not whether he is against BP sponsorship. Jeremy Deller, the former Tate trustee who will represent Britain at this year's Venice biennale, says "sponsorship has always bothered me but you have to realise that this is not a novelty, art has always had connections to power, business and politics".

"No cool organisation, no organisation that does a great thing in the world, will ever sponsor the arts because they don't need to. And so arts sponsorship is always going to be linked to organisations that are problematic. The good thing about BP is at least we see it: the damage wreaked by the Deepwater Horizon crisis could not be clearer. You don't really see the effects of the banking crisis in the same way. As long as people are happy to drive their cars, they can't really complain."

Where all are agreed is on the necessity to ensure sponsors do not call the shots. But while Deller says they "have absolutely no say" with regards to content, others point to more subtle forms of influence. "We will never have conclusive evidence of what has not been programmed, who has not been invited, which work has not been made," says Platform's Kevin Smith. He points to events at the University of Wyoming a year ago, when an installation by British sculptor Chris Drury – Carbon Sink: What Goes Around Comes Around – featuring blackened wood and lumps of coal, was removed following complaints from politicians and mining industry donors.

Walking around Tate Modern, its open architecture a testament to its commitment to the principle of public access, its transparent collection boxes bearing the modest request "Come back soon please donate £4", its galleries filled with political works of all kinds, it is extremely difficult to imagine anything like this kind of pressure being brought to bear here.

Activists say this is not the point, that the question is not the role sponsors play in museums, but the role arts institutions play in lending companies a socially responsible sheen. And they remind me that Liberate Tate came into being after curators emailed the organisers of a workshop they had commissioned from the Laboratory of Insurrectionary Imagination with the instruction "to be aware that we cannot host any activism directed against Tate and its sponsors".