This practice of placing dots after contractions of titles such as Mr and Dr had its origin in handwriting and printing conventions. It is not really a grammar issue. In the 17th century or thereabouts, the dots were called suspension marks, and were placed under, not after, the (small) last letter of the contraction. Later it was considered more convenient to place them after, and since approximately the end of World War II, in British English at least, it is considered convenient to leave them out altogether. Such is progress.
Regarding initialisms, full stops are somewhat more often placed after each initial in American English (e.g., U.S., U.S.S.R.) than in British English (e.g., US, USSR); however, for acronyms that are pronounced like words (e.g., NATO), full stops are omitted in American English.
Posted by: BostonDiner
I remember somewhere seeing the "rule" about not putting in full stops at the end of abbreviations which would end in that letter if written in full. This would have been a British interpretation.
This was a British import, now largely disregarded, of the French rule that a period only follows an abbreviation if the last letter in the abbreviation is not the last letter of its antecedent: "M." is the abbreviation for "monsieur" while "Mme" is that for "madame".
Some people follow a rule that an abbreviation, a word shortened by cutting off its end part, takes a period, but a contraction, a word with the middle left out, does not. Thus Dr James Smith Jr studied Eng. Lit. at college.