Of course, a poem may be interpreted however the reader wishes. My interpretation does not agree with that of the first answer.
Here are the lines about this age of life
And then the lover,
Sighing like furnace, with a woeful ballad
Made to his mistress' eyebrow.
To me, this is meant to show how ridiculous people can be when in love. I think that the reason for the "eyebrow" is that people in love think their lover is so perfect that they will write poems praising even such body parts as an eyebrow.
The reason, to me, that he sighs "like furnace" is because he is so overcome with passion. He is burning and sucking up all the oxygen around him. So I think the description of this age is about the foolishness and passion of people who are young.
The way in which Shakespeare describes the character is with a lack of interest or sentiment towards his mistress, as he doesn't even have a desire to look in her eyes.
Shakespeare's poem reflects the self-reproach or remorse one feels through the great sense of guilt that one has acquired through the infancy and childhood stages that came before this early adulthood stage.
This may be a result of a loss of love. In order for the guilt to be manifested, man uses poetry, music or some other form of culture as a means of expressing oneself.
pohnpei397 is correct here.
The furnace reference has to do with heat, such as that heat which is closely associated with sexual passion. Shakespeare is using the furnace not just for the sighing sound that it makes, but to infer that the lover is "hot" with love for his mistress. It's a great simile that is ahead of its time from a literary standpoint.