Obama already knows that such an event would create an economic drag in the next year of nearly 4% as the various taxes and mandated spending cuts sap economic strength. After four years of effort, bailouts, incentives and programs to keep the economy afloat - what incentive would there be to willingly go over the "cliff?" It is an interesting question.
According to the American Council For Capital Formation  here are the following impacts to the overall economy:
Real GDP- Increasing the current capital gains and dividend tax rates shows noticeable negative effects on the U.S. economy compared to the Baseline in the shorter run. In this simulation, real GDP growth decreases 0.1%, on average, per year, which equates to a $79.2 billion decrease per year over the 2013-17 time period. The results are similar in longer time period: Between 2013 and 2021 period, real GDP decreases $80 billion on average, per year.
Consumption Spending- Consumption spending is also weaker, averaging $155 billion lower per year between 2013-2021. Between 2013 and 2017 time period, the decrease in consumption is a little over $122 billion.
Employment- In the capital gains and dividend tax increase simulation, the job impact is worse between 2013 and 2017 period. The economy ends up losing 380,000 jobs on average per year. In the longer period, 2013-21, the loss is 344,000 per year. Nonfarm payroll jobs show a large loss of 561,000 persons in 2015 and then smaller losses in subsequent years.
Capital Spending- Spending for business investment declines when tax rates on capital gains and dividends revert back to pre-Bush levels; on average $20 billion yearly between 2013 and 2021. The decrease is smaller for the shorter term, 2013-17, $17.9 which is 1.1% lower than baseline.
Savings- At a higher tax rate on capital gains and dividends, saving decreases according to the Flow of Funds by an average of 0.1 percentage points per year in the 2013-2021 period.
Financial Indicators- Both the S&P 500 Price Index, and S&P 500 Earning per Share, decline when the top tax rates on qualifying dividends and capital gains are increased compared to the Baseline. The index declines by an average of 16% and the S&P 500 Operating EPS is down an average of $1.6 over 2013-17. Between 2013 and 2021, the index declines by an average of 14.5% and the S&P 500 Operating EPS is down an average of $2.
Federal Government Budget Receipts- Higher taxes on capital gains and dividends significantly harm the economy and job growth and suggest that the increase in federal tax receipts may not be a worthwhile tradeoff. Despite all that damage to the economy, the overall impact on the budget deficit is only $7.3 billion annually between 2013-2017 and $70.4 billion between 2013-2021 when the dynamic effects on economic activity and induced decreases in tax receipts from the higher tax rates are reflected.