We are not addressing the general question of the desirability of avoiding internet advertisements, some of which are not appropriate. Hopefully, ads on a religious site are appropriate.
At first glance, your question seems to depend on the broader question of intellectual property rights, which means the following. Reuven produces something of value to the masses and, due to financial considerations, attaches conditions to its use even for those who pay for the item. The most prevalent such condition is forbidding someone from copying the content (e.g., of a book) and passing it on to others. In your case, the site owner, who spent money on and would like to profit from it, wants to make use of the site and its content conditional on your allowing advertisements on your screen, because he gets paid according to the number of people who see the advertisements.
Much has been written on the topic of intellectual property rights.[2] Our opinion[3] is that there is halachic basis for guaranteeing such abstract rights in various contexts. There are three main halachic claims to uphold copyright rights, although none of them is both unanimously held and applicable in all circumstances:[4] 1) It is a form of theft, which can apply even though no physical object is taken. 2) There is a requirement to pay for benefit received from another’s property (neheneh), which the copyright abuser does not plan to do. 3) Dina d’malchuta dina – the law of the land upholds many of the creators’ claims to ownership.
Ad-blocking causes great losses to many website owners. On the other hand, historically, most technological innovations, including the internet itself, have enriched some and impoverished others. Furthermore, the legal and halachic logic behind intellectual property rights does not support every claim by every “owner.” We will analyze your case through the prism of the three claims in support of such rights listed above.
Let us start with #3. As far as we have seen,[5] ad-blocking is not illegal,[6] and thus it is unlikely that the website’s warning has any legal standing. Thus, it is very questionable whether dina d’malchuta will forbid using the site with an ad-blocker, which may be why the site warns using religious terminology.
Neheneh is complex to apply in this case. We rule that in a case of zeh neheneh v’zeh lo chaser (the user gains without the owner losing), the user is generally exempt from payment.[7] However, if the owner incurs any damage from the usage, the user has to pay for the entire benefit he received.[8] In your case, the site owner apparently sustains a loss when people use the site, in that it slows down the server, making the site less attractive to those who bring in revenue. While each individual person’s impact is negligible, the owner can argue that he treats ad-block users as a group he is unwilling to allow “for free.” On the other hand, it is possible that users cause more gain to the site owner than damage even with ad-block (analysis of the commercial logic is beyond our expertise), but the site owner wants to pressure the users to provide even greater advantage. If the user indeed provides benefit for the owner, in addition to the benefits that the user himself receives from the arrangement, there is no neheneh obligation.[9] If the content the user is seeking is readily available on other sites that do not make demands on the user, there may not be enough user benefit to warrant pay in the first place. However, it is difficult to be conclusive on this matter. Regarding #1, it is also far from clear that simply entering an open site and reading the content could be considered stealing its content.
Although we are not confident that the above issues make it forbidden to use ad-blocking software against the wishes of the website, we still believe that the owner can make it forbidden to use the site. Even in a case of zeh neheneh v’zeh lo chaser, if the owner says up front that he forbids usage without payment, it is forbidden for someone else to use his property even if the owner loses nothing.[10] At first glance, this restriction applies in your case only according to those (far from unanimous) opinions that maintain that intellectual property is “owned,” such that the prohibition of stealing applies. However, here the owner is in a stronger halachic position than, say, the composer of music. This is because the user is connecting to a physical server, owned by the site owner or, more likely, by a web host that he pays for its services. Therefore, usage is like using a remote control to use someone else’s equipment against his will, which is forbidden.
Based on the above, our tentative position in the standard case of a site that protests against the ad-blocker is that an owner has the halachic power to forbid the ad-blocker to use his site. (What one can surmise from a situation in which the site can prevent access, yet chooses not to do so and just writes that it is forbidden, is a pertinent question that we are not going to attempt to determine.)