tl;dr: Your reasoning that taxes should not increase GDP by themselves as they do not constitute any new production is on right track but might not hold always. The two main reasons for that are:
If we talk about GDP as the actual statistical measure not just as a theoretical measure we run into practical problems when we try to measure it. The most relevant one for this question is the fact that GDP cannot capture all production as statistical offices have no way how to record non-market transactions but at the same time government can still tax even non-market activity and thus increasing GDP in this way by 'revealing' previously hidden production.
Taxing production of some people and transferring the proceeds to other can in short-run increase output and thus also GDP during recession. This is because consumer spending affects GDP not just directly but also has a multiplier effect and during recession transferring money from people with low marginal propensity to consume to people with high one through taxes actually can increase GDP as spending of those people has higher multiplier attached to it.
Full Answer:
In a closed economy, the GDP/output $(Y)$ under expenditure approach will be given by
$$Y= C + I + G$$
where $C$ is consumption, $I$ investment and $G$ gov. spending. You are completely correct that taxes affect consumption. In fact this is made clear in any undergraduate textbook (such as Blanchard et al Macroeconomics: an European perspective). Following, Blanchard et al. we can explicitly make consumption to be a function of output and taxes and in fact we can write:
$$Y = C(Y-T) + I + G$$
where, $T$ stands for taxes. So GDP is function of taxes but at the same time it is function of output itself so in order to analyze the effect of any part of the equation on output we first have to solve the model for output. In order to do so explicitly and directly we have to make some assumption on the consumption function. A standard textbook one would be just simple linear consumption $C= c_0 +c_1(Y-T)$, where $c_0$ is an autonomous spending - spending that does not depend on income/output/GDP (these are macroeconomically all the same so I will be using them interchangeably when referring to $Y$) and $c_1$ is the marginal propensity to consume people have (for example if people consume $75\%$ of their disposable income and save the rest $c_1=0.75$).
By substituting the consumption function for $C$ one might solve for $Y$ to get:
$$Y = \frac{1}{1-c_1}\left( c_0 + I + G - c_1 T\right)$$
Now, the result above shows, that if all else is held constant if you increase taxes the output would decrease by amount $\frac{c_1}{1-c_1} T$.
Of course, assumption that everything else stays constant is not reasonable here. If we make a reasonable assumption that government will use those taxes to fund its spending which we can do by assuming balanced budget $G=T$ the taxes will have no effect on output as their effect cancels each other out. To see this substitute $G=T$ into the expression, that gives you:
$$Y= \frac{1}{1-c_1}\left( c_0 + I + T- c_1 T\right) \implies Y= \frac{1}{1-c_1}\left( c_0 + I \right) +T $$
In this case the increase in taxes would have no extra effect on output/GDP. The output would increase by amount of $T$ but since $T$ is subtracted from the output/income in the first place the GDP would remain at the same level as it was before changes in taxes. (Note, the same would occur if we would not hold $I$ constant and allow for public saving since $I=S+T-G$ - we would arrive at the same conclusion but in more convoluted way).
This being said there are some caveats here. GDP is an imperfect measure so you might observe that the above wont always hold if you would just look at GDP statistics. The analysis above assumes GDP can capture all output. However, that is clearly not correct. If you have home garden and produce radishes for your own consumption, clearly such consumption wont be counted on national accounts. However, now if government is able to somehow levy a tax on output of your radish garden, in one way or another, GDP statistics might actually increase because the portion of your output that was previously 'concealed' from national accounts statistics would suddenly become recorded. However, this rather reflects the flaws of GDP as a measure and practical data collection issues than any economic effect.
In fact because we cannot perfectly observe all output and its components you will routinely find discrepancies between trying to calculate GDP using income, production or consumption approach even though they are by construction all equivalent to each other.
Moreover, even more important caveat is that the analysis above assumes that all people have the same marginal propensity to consume $c_1$. This most likely does not hold in practice and taxing people with low marginal propensity to consume and transferring it to consumers with high marginal propensity to consume would actually be able to increase GDP. However, this occurs only in recessions as in normal times saving would just increase $I$ but during recession periods that does not generally hold.
There are some parts in your question that are also not correct that I would like to just briefly address:
My thinking is that the profit of the producer would not increase, since the price increase on the goods would just offset the cost of the tax.
This is not correct because it assumes that any tax is fully passed on to consumers - that is not accurate in most situations. However, when you calculate GDP using income approach you actually account for taxes in the calculation.
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For households that do save, they either lessen their savings or spend less on other stuff. And savings is just future consumption, meaning that if they save less, they have less consumption in the future.
This is correct but note future consumption is recorded in future GDP calculation not in the present one. However, this would be reflected on present period GDP through changes in $I$ as dis-saving is negative investment since investment must be equal to private + public savings.
