(In contrast to the accepted answer) Yes, you can move the point of origin for the spell effect: specific beats general.
It is a general guideline that specific rules override general rules in D&D. The general rule is that spells require line of sight and line of effect. However, both of these spells have specific capabilities that override those general rules.
A 5-foot diameter sphere of fire appears in an unoccupied space of your choice within range.
As a bonus action, you can move the sphere up to 30 feet.
A silvery beam of pale light shines down in a 5-foot radius, 40-foot-high cylinder centered on a point within range.
On each of your turns after you cast this spell, you can use an action to move the beam 60 feet in any direction.
Both spells note no line of sight is required.
Both note that you can move their effects.
The spell Arcane Eye supports this interpretation.
The spell Arcane Eye wouldn't work as it is described in the spell if this interpretation were not true. Arcane Eye describes passing through one inch holes and out the range of sight (either leading out of line of sight or effect). These spells use the same type of language to describe their capabilities as Arcane Eye.
You create an Invisible, magical eye within range that hovers in the air
As an action, you can move the eye up to 30 feet in any direction.
If Arcane Eye were limited to Line of Effect and you had to see the destination, then it could no longer be used to scout areas the moment you couldn't see it.
Overly Technical Note #1: One might try to argue that the phrase "there is no limit how far away from you the eye could move" is what allows the eye to move behind barriers. That argument could be challenged with the scenario of an eye moving along the radius of a circle around the player which keeps the eye neither closer nor further away as it moves. DMs embracing that interpretation would have to reject the ability of the eye moving along that path if it goes behind an obstruction because the eye is no further away from the player.
Overly Technical Note #2: One might try to argue that because the Arcane Eye can "see" the destination that this creates a Line of Effect. This is contradicted by precedent as per answers to this question: a spell's Line of Effect does not require Line of Sight unless the spell says so. A spell does, however, require an unobstructed path to the target to have a Line of Effect. In this episode of the Official D&D Podcast, Jeremy Crawford clarifies these rules (at ~34 minutes) by specifically mentioning that a closed, but transparent, window blocks casting. Thus, though a transparent window may give you Line of Sight through the window - you don't have Line of Effect because you do not have an unobstructed line to the target and the result is that the spell is blocked. Thus a spell like Arcane Eye does not gain Line of Effect because it has Line of Sight: it gains Line of Effect because the spell says it does. If DMs stayed with the general Line of Effect arguments for spells, then Arcane Eye would fail. But DMs generally don't make it fail because "specific overrides general." Arcane Eye gains Line of Effect because the spell says it does using the same type of language as Flaming sphere and Moonbeam.
Jeremy Crawford's unofficial tweet supports this interpretation.
Crawford clearly notes:
The text of the flaming sphere spell explains how the sphere interacts with barriers, creatures, and pits as it rolls around. As the caster, you don't have to see where you move it –January 25, 2018.