The rule in question is imposed by the IRC Sec. 7872. You would probably be better off with a professional tax adviser (EA/CPA licensed in your State), but this is how I read it:
If the loan is less than $10K - don't worry about it, no-one cares, unless you used it to buy an income-producing property (not your case).
If the loan is less than $100K and you don't have net investment income (see the note below) - don't worry about it, no imputed interest is calculated. This may be your case, depending on your investments. Note, I disagree with Dilip on the interpretation of this part: the investment income, IMHO, is your total investment income, regardless of the income-producing property (if any) you purchased using this loan. This rule is to avoid shifting income from higher brackets to lower brackets.
If there's net investment income - the lower of the net investment income or the calculated imputed interest is accounted as interest. This is codified in Sec. 7872(d)1.A.
If the loan is above $100K - calculate imputed interest and it all is accounted.
If there's imputed interest - it is considered a gift, and if it exceeds $14K (together with all the other gifts from that person to you during the year) - gift tax applies and gift tax return should be filed.
All the accounted interest should appear on your uncle's Schedule B as income to him.
Note: the amounts are aggregate, not per loan. I.e.: if your uncle gave you $70K now, but another $31K last year - the amount in question is not $70K, it is $101K.
Another note: if your net investment income is less than $1000, then it is considered to be $0 (as you mentioned in your answer). This is called "de minimis rule", and is codified in Sec. 7872(d)1.E(ii).
To answer your specific questions, as it seems you've got the Sec. 7872 conditions right:
Does the fact that I used the loan to pay off a student loan affect things?
I drew it up as a "demand" loan - so that means he only has to pay taxes based on the short-term AFR, right? So for 2014 that was about 0.34%, so the income is only $238. But it's supposed to be compounded semi-anually. Did I do that correctly?
What sort of document should I send to my uncle indicating how much he owes in taxes? I'd like it to be good in case the IRS ever looks.
Write him an email saying "dear uncle, my investment income this year exceeds $1000 limit, so you're on the hook for the imputed interest. Sorry, your loving nephew".