Update- The below answer was based on the tweet being about sales tax, i.e. the Amazon tax issue. However it appears that Trump's also talked about another tax topic according to "Amazon.com wins $1.5 billion tax dispute over IRS", Reuters.
The Washington Post has published editorials in support of online taxes. The most direct editorial that I can find is from 2012-07-15:
Virginia voted this year to close the so-called “Amazon loophole,” which allows the online retailer to avoid paying the same local sales taxes that brick-and-mortar establishments charge. California and Texas have taken steps to do the same.
The change is overdue. There’s no reason to favor e-commerce at the expense of other business, and it makes no sense to require some businesses to charge more for the same goods. To that end, a bipartisan bill under consideration in the Senate, the Marketplace Fairness Act, would grant states the authority to require “remote sellers” to collect sales tax in exactly the same way as local businesses are already made to do.
-"Taxing online purchases", The Washington Post (2012-07-15; emphasis added)
However, Jeff Bezos's plan to buy the Washington Post wasn't announced until 2013-08-05.
Since that time, most of the articles I can find are factual rather than advocating an opinion. One of the most directly related is from 2016-02-05, in which the Washington Post reports:
Amazon has long lobbied for the creation of a national standard for collecting state sales tax for online purchases, a change that would simplify their business practices.
In a recent town hall meeting with Washington Post employees, Amazon chief executive Jeffrey P. Bezos -- who also owns The Post -- cited state sales taxes as the lobbying issue he has been personally most engaged on. But so far, legislation at the heart of that change has been unsuccessful -- much to Bezos' chagrin.
-"Why Amazon is doubling down on lobbying", The Washington Post (2016-02-05)
Here, they report that Bezos wanted a standard for tax collection, though don't seem to offer an editorial opinion on it.
Is Fake News Washington Post being used as a lobbyist weapon against Congress to keep Politicians from looking into Amazon no-tax monopoly?
It's unclear what he was talking about since Amazon.com seems to be already collecting sales tax in addition to having lobbied for online tax regulations for years.
The context appears to be H.R.2887: "No Regulation Without Representation Act of 2017", which is cosponsored by 8 Republicans and 1 Democrat. It appears to restrict how sales taxes can be collected. FreedomWorks, a conservative advocacy group, has posted a strong endorsement:
On behalf of our activist community, I urge you to contact your representative and urge him or her to cosponsor the No Regulation Without Representation Act, H.R. 2887, introduced by Rep. James Sensenbrenner (R-Wis.). The bill would codify a 1992 Supreme Court decision preventing cross-border taxation when a business does not have a physical location in which the consumer resides. [...]
The Supreme Court sided with Quill and decided that since it had no physical presence in that state, it did not have to be subject to that tax. The decision in the case has protected Internet-based commerce, prevented revenue-hungry states from taxing sales from out-of-state businesses or merchants.
-"Support the No Regulation Without Representation Act, H.R. 2887"
Since Amazon.com is already collecting sales tax and appears to be for creating stronger tax regulations, while Trump's party is advocating a reduction in tax collection (as interpreted by FreedomWorks), Trump's claim doesn't appear to correspond with reality.