Are there any principles or guidelines we can follow in handling what appear to us as non-rational objections to the gospel?We want to start by saying a few things generally about the whole domain.
To begin this task we will assume that in one way or another we are presented by objections to the gospel. Sometimes the issues are explicit because they are clearly articulated or written and other times they are implicitly conveyed by more opaque and 
interpretable means. In the latter case one of the main ways objections are conveyed, especially in conversations, is by means of body language or what some call micro-behavior.
So especially when we are presented with objections (or potential objections) in the latter way it requires us to properly decode the subtlety of body language in order to correctly discern an objection and its nature. So step one involves us being attentive enough to read and discern a potential objection through what a person says (or writes) and how it is said.
Assuming there is a basis to believe there is an objection to the gospel (and not just an “objection” to ourselves or how we are saying what we are saying) we need to discern the nature of the objection. Is the presenting objection one of a rational nature of the sort one might face when persuading a person to see the rationality of modus ponens or modus tolens? Or is it of the discernible kind that is more influenced by certain kinds of emotional bias?
This discernment is probably the most fundamental issue to resolve in any persuasive conversation.
The sort of questions you need to ask yourself about both written, spoken and non-verbal objections are what can you learn about the nature (rational or non-rational here) from the way the question was asked or presented. Was it expressed with an inappropriate amount of emotion (more or less) or what seems to be inappropriate micro behavior? These are the sort of tips (some people call these sorts of things “tells”) that can inform you if the objections are of a non-rational nature or have a non-rational component to a rational question. The thing to see is that YOU DO NOT WANT TO TREAT A NON-RATIONAL OBJECTION THE SAME WAY YOU TREAT A RATIONAL OBJECTION.
The resources in this section is then aimed to help you make that subjective interpretation with greater precision and accuracy. And, if and when a certain kind of non-rational objection surfaces knowing the most probabilistic successful way to handle it. Other tabs in this section are aimed to help you discern and successfully “treat” non-rational objection. However, remember that what we are speaking of here in terms of “successfully” handling the objection does not mean that you can manage their responses. It means that you have done appropriate things to help move the conversation along if the person with whom you are engaged wants to do so.