No, it's not illegal...
Ads are shown as a contract between the site that hosts it and the advertising company. The contract does not stipulate that customers need to buy something, in fact, the contract can't force the customer to buy anything! At best, the contract can pay the hosting site based on the ad being shown, clicked, or any sale made after it.
...but you will do exactly the opposite
Advertizement is made to put your own brand into the head of people. In fact, most ads don't have any effect on people. As I am writing this, ads for a kid TV, travels to Turkey and the primetime films for the weekend on the TV-station I have on are shown. I have no intention of consuming any of these advertised products.
Impact of a campaign is measured by two metrics: people reached, and people responding. People reached is measured in clicks. People responding is measured in changes in earnings or sales. Clicking the ad increases the reached rating. If the rating is bad, the ad campaign is just ineffective... which leads to the most paradoxical thing: Bad advertisement and good advertisement both lead to more advertisement - bad to level out the missing response, good to maximize the response. By clicking on the ad you just funnel more money to the ad industry.
When does it get illegal? [DDOS]
The only way it would become illegal is if John Doe sets up a computer - or rather a botnet - and has that network click the ads thousands of times per second. Google can handle easily 83,000 searches per second, twitter gets more than 9000 tweets that are distributed to millions of people, Tumbler and Instagram handle together about 2500 posts per second. In fact, every second, more than 100000 Gigabytes of traffic run through the net. To have an impact on one site, you need to be truly a large number of calls... and then it is called a DDOS. DDOS is illegal under the CFAA, in this case 18 UC 1030:
(a)Whoever— (5) (A)knowingly causes the transmission of a program, information, code, or command, and as a result of such conduct, intentionally causes damage without authorization, to a protected computer;
(b)Whoever conspires to commit or attempts to commit an offense under subsection (a) of this section shall be punished as provided in subsection (c) of this section.
In the UK, you'd break the Computer Misuse Act of 1990 section 3, because denying someone else service via DDOS is unauthorized, clearly unauthorized, and prevents access to any data (the website) on any computer (the server):
(1)A person is guilty of an offence if—
(a)he does any unauthorised act in relation to a computer;
(b)at the time when he does the act he knows that it is unauthorised; and
(c)either subsection (2) or subsection (3) below applies.
(2)This subsection applies if the person intends by doing the act—
(b)to prevent or hinder access to any program or data held in any computer
But can a DDOS be protest? [NO!]
Anonymous attempted to petition to make DDOS a legal form of protest in 2013. The petition got 6,048 of the 25000 signatures needed to warrant an answer by the white house - unlike people in 2016 asking for a Death Star.
At least it prompted Joshua I. James to write a research paper about the proposal in March. He too points to the CFAA and Section 5A, especially the sentence I quoted above. Among a lot of stipulations, he points out that internet protest in the shape of a DDOS would need to follow the same rules as a legal protest on the streets - which for example demands that entrance to businesses can't be blocked, and one is not allowed to harass employees and customers.
According to the general rules for legal protest as given, there are still a number of challenges. First and foremost, entrances to businesses should not be blocked. In terms of DDoS, if sustained denial of service takes place, then access (entrance) to the server (business)is eﬀectively blocked. This means that, at a minimum, sustained denial of service should be considered as a non-legal approach to protesting.
Thus, he concludes sustained DDOS is per se can't be a legal protest, and even a non-sustained DDOS would impact people using the site in a way they will deem harassing - which means that even a non-permanent DDOS can't be a legal protest.
And then comes the final blow: DDOS, unlike a real protest on the streets, can't, by its very nature, inform people of why there is protest, even if it were a form of protest! This means that nobody knows it is meant to be a protest and not a normal DDOS, and as it can't convey what the action is about, it can't be a proper protest.