Can I have a clothes dryer and room heater on one circuit?
Get a 120V heater

Electric heaters are cheap.  A 240V 1000W heater is worth about $40.   You could easily spend more than that on exotic breakers and hookups.  Sell it on Craigslist and get a 120V 1000W heater.  

If you hardwire it, either put it on a dedicated 15A circuit, or a shared 20A circuit. A hardwired load which shares a circuit with receptacles must be less than 50% of the circuit's capacity (2400W, which is why you can't do it on a 15A 1800W circuit unless you downsize the heater.) 

Share a 40A breaker

No.  You are not allowed to put more than one receptacle on a branch circuit.  NEC 210.23.

To be more precise, 210.23 calls out the exceptions where you are allowed to have multiple receptacles on a circuit, and it lists them by ampere rating. For residential, only cooking appliances for a 40-50A circuit.

Each receptacle must have the same rating as the circuit, i.e. you can't put a 30A receptacle on a 40A circuit.  (NEC 210.21).  The exceptions are 15A receptacles are allowed on 20A circuits... and 50A receptacles on 40A circuits (because no 40A receptacle standard exists).

Duplex breakers

The usual way to solve a "box cram" problem is a quadriplex breaker that puts two 240V 2-pole breakers in two spaces.  Quadriplex may not be available in 3/4" wide breakers.  But you don‘t need them to be.   You can get two plain 120V duplex breakers and use them for four of your 120V circuits.  That will free up two spaces for your 2-pole breaker.  

Sub-panel

Given the criterion of using the on-hand parts (240V/40A breaker and heater), sub-panel is the way to go. Put in a 240V 40A sub-panel and put in any circuits you please, does not necessarily have to include the dryer. In fact I wouldn't put the dryer in there. I'd move over as many 15-20A 120V circuits as I possibly could, to free up spaces in your main panel. You could certainly put at least six 15A circuits - I'm not familiar with the rules on panel fill. You could even put this sub-panel right next to your main panel to make it easier to move circuits.

Do not be bashful about buying a large sub-panel.  Cheaping out and getting a panel with too few spaces is how you got in this pickle in the first place.  You might even think about planning the sub-panel so it could become your future main panel.  

It is totally OK to buy a sub-panel which is rated at a larger amperage than you are using.    It is also OK if your sub-panel has a "main" breaker larger than your 40A  breaker.  That main breaker will simply be redundant.