The Scriptures that come to mind are Acts 2 regarding the members of the church sharing with others members.
The only "collection" that I can recall from Scripture is the collection that was taken for the poor saints in Jerusalem, which was a specific need and situation.
It does not appear that there was a weekly "offering" in the early churches for some general purposes ultimately unknown to the members.
Probably one of the main reasons churches today have resorted to a weekly "offering" is because they have placed unneeded monetary overhead on themselves in constructing large buildings, which cost quite a bit of money to maintain. Therefore, they are forced to ask for money from the members on a weekly basis to even stay afloat.
I submit this is unwise, and could have been avoided by "splitting" the church into geographical house churches led by small groups of sound men in the church. Then, perhaps the large group could rent a location once a month (or two) and come together for fellowship, etc.
If we consider that the early church met in homes, and that elders who labored in teaching the Scriptures could have been provided for by food, shelter, etc. rather than cash, it really seems there would have been no need for a weekly "offering".
Regarding your original question, I think we should support the church we attend. However, perhaps it would be wise to respectfully ask the elders if maintaining a building is the wisest use of our money and perhaps suggest other more Biblical models.
Grace to you,
Taylor