No, it regularly does not. But when you are currently in an apprenticeship, you get a Duldung to finish the training. That is why we always recommend to start as soon as possible to look for an opportunity for an apprenticeship.
Another interesting point is the ‘3+2-rule’: If you finished a 3 year apprenticeship and you stay in the same company, you are getting a Duldung for the next two years as well. Otherwise you have 6 months to find another work in your field.
Having a job or not does not influence the asylum seeking process because the only thing that matters in this context are reasons for your flight. But in case you got an unlimited work contract and you can prove that your German is on C1-level after three years than you can apply for a Niederlassungserlaubnis (permanent residency permit) which grants you the right to stay in Germany.
For people that received a negative result on their asylum seeking process it does make a difference if they start an Ausbildung or not: if you are doing an apprenticeship than you can stay at least until you finished this training plus if you get employed after you succesfully finished your Ausbildung, then you will receive the possibility of a residency permit to work.
When you have a Duldung you can only stay for the time of the training and if you have a job in the same field you can stay on two more years. But in the end you may have to leave the country after a maximum of five years. In 2006 there was a case where the government decided that people with a perennial Duldung can stay, but this is an exception from the rule.
Taking up a job or an apprenticeship might not influence the question if you get asylum or not, but when the situation in your home country changes and you might have to go back, it will be taken into account how well you're integrated into the German society - for example with your job.