The CRC is the source of a detailed article (written January 2008) that explains the reasoning behind the discussion of the generation of carbon dioxide. The significant part is at the end where they say (which I put in bold) that they have decided to be strict on the matter because it involves chametz on Pesach. They only certify CO2 that is definitely not derived from Chametz even though it may not be required to do so. Rav Moshe Heinemann of the Star-K has ruled that it is not required.
According to a footnote, the OU would rely on a letter from the supplier as to the source of CO2. The quote from the OU by @GershonGold would seem to imply that they might no longer require a letter from the CO2 supplier, but that is not explicit.
The footnotes give details as to how CO2 was derived, the psakim over the years, and the methods as they changed and became technologically advanced.
There is a discussion at Carbon Dioxide for Pesach which explains how CO2 generation was originally chametz, but nowadays is no longer a problem.
With this understanding, we can now address the question of whether
carbon dioxide recovered from a chametz fermentation should be
forbidden on Pesach. It turns out that the technology of recovering
carbon dioxide has steadily progressed over the past 100 years. At
first scientists figured out how to liquefy the carbon dioxide
escaping from these reactions but had no effective method of purifying
these vapors. As such, in those days, the liquid carbon dioxide
likely did have a carryover of taste from the beer or whisky it was
made from, and for that reason, the Poskim of that era concluded that
the carbon dioxide was forbidden on Pesach.
However, in the past 50 years, the industry has developed 5 different
methods of purifying the carbon dioxide – four while it is in the
gaseous state, and a fifth after it is liquefied – to the point that
nowadays carbon dioxide recovered from beer or whisky bears absolutely
no taste of the original liquid it was created from. We can surmise
that that this change of facts is why when the question was presented
to Rav Auerbach, he ruled that it is surely permitted! Rav Auerbach
appears to have held that a vapor/condensate only retains the status
of the original liquid if it also retains the taste of that liquid,
and since nowadays the carbon dioxide has absolutely no taste of the
original beverage, it is not forbidden as chametz. [See the footnote
for an alternate explanation of these halachos.]
Of the kashrus agencies we conferred with, we found that many chose to
take a strict approach and not allow carbon dioxide from chametz or
even kitnios (but they took varying approaches to verifying the source
of the carbon dioxide) but others came to the conclusion outlined
above that one may be lenient nowadays. (This is the ruling of Rav Moshe Heinemann of the Star-K). Due to the (commendable)
strictness with which people traditionally treat questions of chametz,
the cRc only certifies seltzer or soda after verifying that the carbon
dioxide is not recovered from chametz (but does accept it from kitnios
sources). While this may not be required on strict halachic grounds,
it is within the spirit of חומרא דפסח to be machmir on this matter.
The following are some relevant facts about the current carbon dioxide
Nowadays, beer companies typically do collect the carbon dioxide
vapors escaping from their fermenters for reuse in their products, and
it is very rare for them to sell the carbon dioxide to others.
Most whisky and ethanol produced in the United States is kitnios, not
chametz. Due to an overabundance of carbon dioxide byproduct, it is
uncommon for a producer (e.g. a whisky or ethanol plant) to recover
the carbon dioxide unless they have a specific buyer in mind. In
fact, typically the carbon dioxide resellers are the ones who install
the collection equipment in the producer of their choice, and then
remain with that producer for many years. As such, although many new
ethanol plants have opened up in the USA in recent years, it may take
many years before the carbon dioxide produced in those plants is ever
used in seltzer.