If a customer asks you to perform tasks/ services that are out of your company contract, and asks you not to inform your company, what would you do?

It is a very sensitive and dangerous situation that can happen quiet offen.
I woudl say that the decision should be initiated by the management of the service provider company.
If I would be the service manager of the company I would say it is depend of the request.
Because although I would liek to keep my customer satisfied but there should be a limit for these kind of "favours".
For example I will not agree to start a project even it is a very small one that is out of the scope of the contract we have.
Why? Because this work is making deficit for my company.
And this is unacceptible from my point of view.

Excellent question! Whatever kind of management one does it's also an exercise of values, moral: transparency, truth, honor, commitment, sacrifice, loyalty etc.; but there also another set of values, technical: profit, quality, durability, cross -selling potential, marketing etc.
It's not easy to change the mindset if you're asked to outcast the first set to achieve results in the latter one...
unless one has slack.
Moral slack means you have political skills, and probably since there so many ways to tell the truth it wont be necessary to lie or hide the truth from management, whom one should be bound by loyalty.
A very experienced professional will not only oversee the incoming concession but also knows how dribble it.
In practical terms if you're developing software the best thing is to leave some empty holes, that way you will see it coming and you know where it will hit the project, like for instance a printing functionality, or an exporting one like export report to excel.
These will not require structural changes, like in the database or in the activity diagram. 
But what if the client asks you to develop an entire data warehouse model to perform business intelligence? That's an entire different story, since it would not be in the project scope, but, don't shut the door and try o understand what the client wants, and suggest some third party software, so you won't be accountable for client's use of it.
There could be many examples: construction, banking, insurance, where one has to develop certain strategies to overcome the alluded situation in the question. 
But the problem could become extremely complex if one is dealing with health matters.
I hope someone could share that experience.
   

It depends on the situation.
Usually I do not prefer to perform tasks/ services that are out of my company contract, but if this task is for the benefit of the company and I know that majority of people would not get affected with this then I will do it.
     

First of all no gold plate, and you must inform your company, if your company agree with that we can do it. 

First you to be sure ( tasks / service ) is not against genral rull or law .
Second you should take permsion form your direct boss or (key person) it is not nesseccery to inform all details but you can give a brief with outline to make what they requied & its cost .
In case you follow the above you make the customer stisfaction without breaking law or rull

Thank you Tamer for your excellent question and also to Francisco for his amazing full-detailed professional answer.
If I would have to add something, I would like to say that as far as the stakeholder satisifications is considered, the client requrements and expectations must be achieved as a commitment and obligation for the sake of the company or the organisation success.The requirements of the client must be bounded by the ethical factors like faithfulness, loyality,honesty,and responsibility.
If the client reguirement is legal , I will manage it in the stakeholder expectation strategy plan and inform my boss with this need to be included in the project plan.
As it is clear from the question the requirement is unlegal because he need not to inform the company, in this case the project manager has to use his personal skills for not violating the ethical values and at the same time not losing the client as a costumer.
Thank you.
I hope my answer is of added value.
 

If what the customer is asking for is "unethical" and conflicts with the terms and conditions of my contract, I wouldn't do it.
It's simply not right and worse can subject you to possible disciplinary action from the company's side.
Customer retention shouldn't come at the expense of the job itself or what is ethically correct.

It is simply get his proposals and transfer them to your boss...he knows what to do....never do thinks without company knows....bad or good .....

if this service can i do them out of work range i will accept  if this services must be done by using compny resources i will refuse 

Hi - first we have to understand the customer's demand if it is legal or illegal, if it is minor than it can be done, or if it affects the reputation of the company going forwrad than we should talk to someone in the company and have second opinion, customer is mandatory no doubt but it is the company and the product that is attracting customer to deal with us, we being employees of the company should consider each and every point first......

Apologize