Finland was (for all intents and purposes) gutted after the valiant effort they provided in the Winter War of 39/40. To expect anything more than what they historically contributed to Barbarossa is folly.
The first part of your comment would be true just after the Winter War, but no longer in the summer of 1941. The Finnish state and military rebounded admirably from the losses of the Winter War, through improving military training and mobilization, and going on a weapon purchase spree. By the beginning of Barbarossa the Finnish Army was a whole different beast from what it was in late 1939 - early 1940. It had more trained men, and a lot more of all kinds of weapons and materiel. It's readiness for offensive operations was hugely improved. During the Winter War, the Finnish Army could not manage successful attacks to save its life (never mind that it was brilliant on the defensive). In comparison, in the early part of the Continuation War the Finnish advance into Soviet Karelia was quite successful all around.
As for the second part of of your comment, I mostly agree. In the early part of the Continuation War, the Finns did all they could. After the advance phase of 1941 was completed, the Finnish Army had shot its bolt logistically and physically speaking. It had no option but to settle on the defensive. Finland was punching above its weight, it was one of the most heavily mobilized societies among all the combatant nations. The country was top-heavy with military, and having a huge percentage of military-aged men in uniform was very costly for the functioning of the national economy itself. In those conditions, the Finns could ill afford further large-scale offensive operations.
Finland was dependent on Germany for food, fuel and other necessaria. But then the Germans could hardly make the Finns attack by withholding food or fuel deliveries, as not only would that raise anti-German feelings among the Finns, it would also directly serve to make the Finnish military less effective in its operations. Practically, then, what the Germans could do would be to try and
bribe the Finns with further resources and goods, to make them commit to further attacks. The Finnish leadership would consider the access to better resources to be the
sine qua non of more offensive action, anyway.
Then there's also the fact that Murmansk and Lapland was strictly the Germans' responsibility. The Finns would not send their troops to attack Murmansk itself. What the Finns could do is attack the Murmansk railway further south, to cut it there. The Finns had a plan for attacking Belomorsk to this effect in 1942. IOTL, this plan was scrapped as Mannerheim feared that launching it would lead to a) the Finnish military suffering the kinds of losses it could not bear (see reasons above) and b) the Western Allies becoming so enraged that the US would declare war against Finland (what it never did IOTL).
Now, if Hitler commits to truly bribing the Finns with different things, he just might be able to convince the Finnish leadership to launch a serious attack against the Murmansk railway on the level of Belomorsk. This would mean not only sending the Finns a lot of military goods and other resources, but also committing several German divisions more to Finland to support the Finnish attack, and to then take Murmansk itself as the Finns would not do any follow-up attacks after (and if) Belomorsk is taken.
So, what I am saying is that even
if the Germans manage to make the Finns attack the Murmansk railway with some real strength, they would still have to commit a lot more resources and troops, moreso than IOTL, into the northern front, to be able to take Murmansk. These resources and men would naturally have to come somewhere.