I have never even heard of stripe, so I highly doubt that they will take over paypal anytime soon. It will take them several years to build up the credibility and connections that paypal has. It is possible eventually they might surpass them, but I doubt it will in the near future.
Stripe won't put PayPal out of business as PayPal is too well establised and used on too many websites. It has a lot of really great features compared to Stripe and is easier to implement and incorporate into a website than Stripe seems too be. People are too afraid to use new payment methods until they have been proven to work
PayPal is too long established and has too much of a hold in the online industry to be put out of business. While Stripe might become a little established itself, it takes a whole lot of market dominance to be able to get rid of a service like PayPal. Twitter and Apple are not enough to do this.
I do not believe Stripe will put PayPal out of business. Why people would want to secure their monies with non-insured services in beyond me. I think the fact that PayPal is in business without these protections should be enough to put them out of business. But none the less Stripe isn't going to hurt them to the point that they close down. Not to mention they're probably "too big to fail".