Using ACS 20012014 in the online query, the HH variable ownershp (tenure) and hhwt (Household weight) I get a table with total weighted N of 4 billion, rather than the expected 116 million. How do I get a table with the correct N?
There is no official data product ACS 20012014. If anything, ACS only started in 2005. Do you mean 2010?
At any rate, some of the reasons to get the wrong population totals are:

You are using personlevel data, so your HH weights are effectively multiplied by the HH size. (You may want to subset to relate==1 or something like that).

You are using annual weights instead of multiple year weights. The multiple year weights are carefully developed by the Census statisticians; there is no guidance on how to use the weights if you combine the years on your own. If you have 10 years worth of combined data, then your annual weights will produce a population total that is about 10 times off.
Finally, without knowing what you did, there is no way for us to tell “how to get the right table”. What software are you using? How did you specify the weights? What were the commands that you typed? Provide a reproducible example.
Using the online analysis system, a weighted tabulation of ownershp will produce a cumulative total for all ACS years (20012014) and will duplicate households with more than one respondent. If you want to see tenure responses by year, you will need to enter “year” as the column. Further, since you are observing a household variable, filtering your results by pernum(1) will make it so each household is counted once. These two changes should help create results more consistent with your anticipated N.
I hope this helps.