should I tell my internship company about this other offer?
Companies generally like when employees disclose information that improves the company's ability to make personnel decisions. Of course, the company's best interests and your best interests may be different.
- Your host company may perceive you as more valuable if they know that another company (perhaps a competitor) also perceives you as valuable.
- Internships are learning experiences; sometimes it is the honest goal of the host company to find their interns great opportunities regardless of whether they stay on with the host company.
- Your team lead may be able to give you some objective insight into what you can expect working at the startup versus what you can expect continuing with their company.
- Some host companies for internships feel entitled to first consideration by their interns, and see applying elsewhere as an indication of disloyalty or lack of commitment. The argument being, "We treat you great, don't you love it here? We don't want someone who doesn't want us."
- Your host company may feel like you're just trying to squeeze them for better compensation after only a few months on the job.
- Even if they're happy for your potential opportunity with this startup, it's not clear that your host company has any way to support you in getting or negotiating that offer. Which leads me to your next question:
Would it benefit me in terms of moving along the negotiations?
Since what's on the table from your host company is only an extension of an internship that you have only been in for a few months this summer, the points in favor of telling them aren't worth much. Ending your term rather than extending it would negatively affect their chances of hiring you into a permanent position later, regardless of whether you have another opportunity in the works. This doesn't sound like a negotiation where very much is on the table for you to begin with, and your team lead is already advocating for you, so I don't see that you have much to gain.
On the other hand, you definitely have something of value to lose, in the support of your team lead. This is the person at your host company who is most likely to feel you owe them some loyalty and consideration in return for their guidance and advocacy. Their personality and how you approach them will make a big difference in how they react.
By all means pursue the opportunity, but in your situation my advice is not to tell your host company until such time as you've been officially offered a position that you would be willing to take. If the offer's not complete then you have nothing of value to tell your host company.
If the offer's complete and acceptable (even if you intend to try negotiating the details) then whether you tell your host company depends on whether they could possibly offer you something better. If not, then the most valuable thing you can tell them is that you're no longer interested in extending the internship—after you accept the other company's offer. But if so, you owe it to your team lead who has advocated for you to let him or her know the details of the offer, including the deadline for accepting or rejecting the offer.