How to review bogus science without hurting feelings?

So, I see that you're conflicted by the desire to give your honest opinion and your desire to be nice. While some say you should let go of the desire to be nice, I think in this case it might be more productive to first look at how you can be nice. But first, let me briefly talk about 'cranks':

What are cranks?

The following quote from Nature is prominently placed on the Wikipedia article on cranks:

A crank is defined as a man who cannot be turned.

Less cryptically, an important property of crank is the unwillingness to change their line of thought or accept being wrong (they likely admit to make 'unimportant' mistakes and will immediately explain what they 'actually' meant). This will become a very useful definition. Suppose for the moment that exactly one of the following is true about the motivation of your correspondent:

1. "I have a brilliant idea about the universe and by showing this to an expert, I'll get the recognition I deserve."
2. "I don't know much about Physics, but I think this is a good theory and want to know what the experts think."

Note that person 1 is a crank, for when you tell (1) that this idea is 'bogus', the reaction will be likely hostile, as (1) will refuse to be 'turned'. Any conversation with (1) is a waste time. If your know for certain that you're dealing with person 1, ignoring, aborting and running away in the most polite way possible is recommended. However, as you desire to be kind to your correpondent, I doubt you are certain that you're dealing with person 1. (Also, assuming 1-ness might be dangerous. Applying Hanlon's razor seems like a good idea.)

How to be nice to person 2

Now, is person 2 a crank? (2) could be a crank, but not necessarily. Perhaps (2) is simply a layman who always had an interest in physics, but never had (or took) the chance to pursue this interest with proper study and thinks this is how physics can be done.

For person 2, I think the best way to be nice is to not give your opinion on the work, other than that it simply is hard for you to judge (2)'s actual ideas as they are very non-standardly presented. (This can be a lie. But I think it is a very useful lie.) It is important that you add the advice that if (2) has an interest in physics, (2) should learn more about physics so that (2) can properly present the theory. It is good to add some explicit method for (2) to do this, such as some introductory books, courses or videos.

In the best case, (2) will start learning things and eventually will realize that his theory is 'bogus' by themselves! In the worst case, (2) will show the inability to 'turned'. But thereby, (2) reveals to be (1) all along and you can therefore safely abort communications with your correspondent.

Conclusion

I think it is good you ask this question, because I believe there is a true dilemma here. You must choose at most one of:

• Share your (brutally) honest opinion with your correspondent.
• Help your correspondent by gently directing them to the path of learning.

Any combination of the two will likely act as a discouragement for your correspondent from attempting to learn, for your correspondents idea that they might be able to do some physics is likely crushed by the weight of an experts opinion.

I believe feelings must be set aside while conducting science.

As @henning said; how would you react to a patient who claims in front of their physician that addiction is not serious (or that cancer can be cured by praying etc.)? I believe that the patient's feelings wouldn't be the priority in that case.

Similar to being a medical doctor, being a scientist is not a hobby. It is a serious business, and should be done rigorously. If a theorem or claim is obviously wrong, bogus, or plain nonsense, best way is to point that out directly:

I think your claim is not true, because you must first prove this and that before arguing about your own theory.

It's not your role to disprove the author's theory; it's the author's role to convince the reviewer of the theory's validity and usefulness.

As StrongBad notes, you should deal with the paper on its own merits.

Handle non-standard interpretations of electrons etc by asking for a literature review that interacts with established theories. Handle non-traditional ideas by asking to see experiment results that differentiate the author's theories from mainstream theories. If it is a purely theoretical paper, ask to see a compare-and-contrast with predictions made by conventional theories. They might not cover all bases, but it is up to the author to demonstrate that their theory is competent - either relative to other theories or standing alone, the latter requiring a lot more to make a convincing argument.

These would not be unreasonable requests on your part. The heavy-lifting needs to be done by the author. Your preliminary role as a reviewer is to check that it has been done.