Answers to clarification questions
There are no "rich people" forms. However, there are forms that are far more likely to be interesting when a rich person files them. For example, when someone claims charitable donations on Schedule A or 8283, they need to list the donations. He also has to write out what personal business related tax deductions he's taking on his Schedule C tax forms. There's also a lot of single use forms. The typical filer only fills out one or two. And of course many of his forms are going to be multiple pages where a typical person only uses part of one page.
There are very few actual disqualifications for President. He needs to be a "natural born citizen", a US resident for at least fourteen years, and at least thirty-five years of age. He can't have been President previously for six or more years. His tax forms won't say anything about his citizenship and are unlikely to contain information suggesting his age is below thirty-five or that he has been President previously. It is conceivable but unlikely that they show that Trump was claiming residency outside the United States. It's far more likely that they show information that would make people not vote for him because they look bad. Such information would not be disqualifying as I would use the word but could still be negative.
Trump is clearly hiding his tax forms. As his sons (Eric and Don, Jr.) have indicated, the main reason for this is that many people won't understand the information he filed. The most charitable reading of this is that his taxes are complex and complicated. A less charitable way of taking this is that his forms show him taking deductions to which he is not entitled or not counting as income things that he should have.
Note that the claim that his taxes have been audited for the last fifteen years suggests that the latter interpretation is correct. The IRS has limited funds for audits. The only way they could justify multiple audits is if the previous ones made them more money than they cost. Trump's sons may believe that this indicates that the IRS behaved inappropriately, but the neutral arbiter of the US court system apparently wasn't supportive.
IRS audits are different from SEC audits or other government audits. Other audits are to detect wrong doing. It's expected for them to cost money and they are budgeted to do so. Tax audits are to increase revenue.
Also note the following from the IRS Audit FAQ:
Generally, the IRS can include returns filed within the last three years in an audit. Additional years can be added if a substantial error is identified. Generally, if a substantial error is identified, the IRS will not go back more than the last six years.
Trump has said that his current audits date back to 2008. That is well over the three year limit, suggesting that either he is getting different treatment or that there was a substantial error on his returns.
Trump less wealthy than he says
Trump has consistently claimed more wealth than other sources think he has. He has put it at $10 billion or more, while Forbes has him at $4.5 billion and Fortune at $3.75 billion. Tax returns would have the exact numbers for both revenue and expense. They also might hint at debt numbers, as they would show interest paid.
Trump overstating charitable contributions
Trump said that he was going to donate to veterans. He failed to do so. He was caught out on it. He eventually made the donation. So, yes, there is good reason to think that he would lie about making charitable donations. Note that he has other explanations for this.
Trump may not pay taxes
Trump did not deny paying zero taxes in the debate. He said that if he was, it was proof of his good business sense. Absent a denial, that seems fully justified. Of course, that would mean that Hillary Clinton's claim that Trump's tax plan would reduce his taxes is bunk, but that's politics. There's no compiler to enforce coherence of positions. She can claim both that Trump isn't paying taxes and that his tax plan would reduce his taxes. And we can make fun of her for being inconsistent the same way we make fun of Trump for buying his suits from China while saying that it is wrong for people to do so.
Appropriateness of criticism
In the tax forms Mitt Romney released in 2012, he actually didn't take some of the deductions to which he was entitled to increase his apparent tax rate. Romney had two big problems in public perception of his tax rate. First, he tithed 10% of his income to religious charities, which was deductible. Second, most of his income was investment income, which gets a privileged rate. The point of this comparison to Romney is that Romney did experience negative results from releasing his taxes. So Trump has reason to believe that his tax returns would be (mis)interpreted in the same way.
It's not clear that Trump is going to have problems with donating too much to charity (too much to have a politically correct tax rate), but a high amount of investment income is very likely. Also, Trump is likely to have a higher business income than Romney. People might use his revenue numbers as a base rather than his income (profit) numbers.
Just as an aside, I personally have no problem with Presidential candidates or anyone else not paying taxes on charitable donations. To my mind, they should be subtracted from the denominator when calculating the tax rate. However, when those numbers are actually calculated currently they do not do this. This has the effect of making those who donate a lot to charity look like they pay a lower tax rate than they really do.
Note that it is entirely possible that Trump's tax returns are clean as a whistle. If so, it's rather silly for him not to show them. Thus, it's plausible to suspect that there is something that either appears wrong (e.g. a low effective tax rate) or which is wrong (e.g. inappropriately taken deductions). So long as he refuses to release his returns, it's reasonable for Clinton to speculate as to why. Similarly, it's reasonable for Trump to speculate as to why she deleted her emails or refuses to release her transcripts.