This article presents a number of charts and data from the Congressional Research Service and other sources that indicates the answer is tax increases do not slow down economic growth, nor do tax cuts increase economic growth.
The nonpartisan Congressional Research Service studied this issue and released a report in September 2012. The author examined individual income tax rates - both marginal and average rates - over the period from World War II to the present. The maximum marginal tax rates range from 70 to 90 % in the 1945 - 1980 period, to today's 35%. Capital gains tax rates were also evaluated, which are also substantially lower today than they have been historically. Pre WWII data were not examined because few Americans were required to file federal tax returns prior to that time.
The tax rates were then correlated with economic growth data from the US Bureau of Economic Analysis. Statistical analysis showed virtually no correlation between tax rates and economic growth. If anything, there is a very slight increase in growth during periods of higher tax rates, though the author notes that this could be a result of other changes in the economy during the time period.
The study concludes:
The results of the analysis suggest that changes over the past 65 years in the top marginal tax rate and the top capital gains tax rate do not appear correlated with economic growth. The reduction in the top tax rates appears to be uncorrelated with saving, investment, and productivity growth.
There is, however, a correlation to income distribution, with lower maximum tax rates leading to greater disparities in income, i.e. greater concentration of income at the very top of the economic scale.