Why is wage-induced inflation a bad thing? More than once, I have heard on the news how the US Federal Reserve would raise interest rates to counter the risk of inflation, even when they indicate that the potential inflation would be caused by "wage pressure".

I can understand why the wealthy disapprove of inflation -- it causes the value of their wealth to decrease -- but for the poor, this is not a significant concern.  And if (for example) the price of products are 50% due to labor, doubling the salary of that labor would only increase their cost by 50%, while the workers who were paid that salary could buy twice as many products. Mathematically speaking, then, I would expect prices to asymptotically approach some new value at a rate of e^(-k/t), not spiral into hyperinflation. 

What am I missing here?  Is inflation due to wage pressure really a bad thing for the general population, or is this a case of the Federal Reserve bowing to the will of those few who would benefit the most?
Wage induced inflation can create inflation spirals, which can lead to hyperinflation. I think everyone agrees hyperinflation is a bad thing for everyone and not just the rich. Furthermore, inflation can be a significant problem for the poor as well.

Such a case of inflation spirals was even threatening post WW2 Germany.

The spiral happens because, if workers demand higher wages that firms cannot pay, then firms may increase prices in order to make up for those losses. This causes inflation, which reduces workers' real wages, so they ask for another wage increase. Firms not in a position to deny or to afford these wages will have to comply and increase prices again. This can go on and on and easily spiral out of control.

In principle it is possible to have higher inflation and higher wage increases. This was the case in Italy before the introduction of the Euro.

However this wage induced inflation also affects non-wage earners, such as those living on welfare benefits, the unemployed living on savings and retirees. 

As for whether fiscal stimulus is better put in the hands of the poor, this is a different topic related to fiscal policy and unrelated to inflation (i.e. monetary policy).